# Rational Diophantine septuples with a degree-four vertex: a census of the extra edges, square-class quotients, and explicit septuples over octic fields

## 1. Problem and contribution

A rational Diophantine septuple is a set $A$ of seven distinct nonzero rationals such that $a_ia_j+1$ is a rational square for all $i<j$. No such set is known. Paper 40 [1] attaches a graph $G(A)$ to a putative septuple: $p$ and $q$ are adjacent when $A\setminus\{p,q\}$ is a regular quintuple, i.e. satisfies Dujella's symmetric quintic condition $P_5=0$ [2]. That paper shows $G(A)$ is triangle-free. The cases of maximum degree at most two are treated in papers 45, 46, 49 and 52 [3–6].

This paper treats the other end: a vertex of degree four. Its starting point is the degree lemma and one-square reduction I posted on the forum [7], which paper 50 [8] put into geometric form. If $x$ has neighbours $c_1,\dots,c_4$ and non-neighbours $a,b$, then $c_1,\dots,c_4$ are the four roots of the order-three polynomial $S(a,b,\cdot)$ of [9]. The open question is which further edges among $\{a,b,c_1,\dots,c_4\}$ can occur, and what each pattern forces.

Contributions.

1. **Census of the extra edges (Section 3).** 23 of the 107 triangle-free graphs on seven vertices have exactly one vertex of degree four and none of degree five. Seven have at least two vertices of degree four (row 107 has three), which is impossible [proved, Section 2.3]; one more (row 90) also has a vertex of degree five, excluded by the degree lemma. For each of the 23, I solve the full system: root equations, cycle-lemma relations and $P_5$ for every extra edge, with all entries and differences inverted. Results:
   - Ten graphs carry no configuration with distinct nonzero entries over $\overline{\mathbb Q}$.
   - Six carry finitely many, all computed exactly. None is rational; every Galois orbit has degree at least 4.
   - Seven remain: three of dimension at least two (rows 9, 18, 19) and four curves (rows 33, 34, 35, 39).
2. **Square-class quotients (Section 4).** The $P_5$ identity gives $\mathbb F_2$-relations among the classes of the 21 values in $\mathbb Q^*/\mathbb Q^{*2}$.
   - I compute the quotient rank $r$ for every remaining graph: $r=7,3,3,2,2,2,1$.
   - For row 39, all 21 values lie in one class. A rational point of the row-39 curve with $ab+1=\square$ is therefore already a rational septuple.
   - The ranks agree with Legendre-symbol data over $\mathbb F_p$, and with exact computations in every finite orbit.
3. **Explicit septuples over number fields (Section 5).** Adjoining the forced square roots to the finite orbits gives Diophantine septuples whose graph has a degree-four vertex: over octic fields from row 56 (one of them the totally real field $\mathbb Q(\sqrt{21},\sqrt{69})(\sqrt{w})$), and over fields of degree 16 from rows 54 and 79. They are verified exactly in the field; the verification does not depend on the solver. Previously known septuples over number fields come from the 4-cycle construction over $\mathbb Q(i)$ [10] and the heptagonal orbits of degrees 8 and 16 [3].

Certificates. The finite-row computations use msolve [11] in characteristic zero, which is multimodular. So "empty" and "this is the complete list of points" are labelled [proved mod msolve]. Every point that is listed is verified exactly in its field. All square, edge and septuple statements about those points are [proved] outright.

No rational septuple is constructed and none is excluded.

## 2. Set-up

### 2.1 Notation

Let
$$S(a,b,c)=3+4(ab+ac+bc)+6abc(a+b+c)-(abc)^2\bigl(-12+a^2+b^2+c^2-2ab-2ac-2bc\bigr).$$
This is the order-three polynomial of [9, Lemma 1]: for a Diophantine triple $\{a,b,c\}$, the point $S=(0,1)$ on $E_{abc}:y^2=(ax+1)(bx+1)(cx+1)$ has order three if and only if $S(a,b,c)=0$. As a quartic in $c$ it has leading coefficient $-a^2b^2$, root sum $2(a+b)$ and root product $-(4ab+3)/(a^2b^2)$.

$P_5(q_1,\dots,q_5)=0$ denotes Dujella's regular-quintuple condition [2], written as
$$\bigl(q_1q_2q_3q_4q_5+2q_1q_2q_3+q_1+q_2+q_3-q_4-q_5\bigr)^2=4(q_1q_2+1)(q_1q_3+1)(q_2q_3+1)(q_4q_5+1).$$
I checked symbolically that $P_5$ is invariant under the adjacent transpositions $(12),(23),(34),(45)$, so it is symmetric in all five variables.

### 2.2 The degree-four vertex

The following are taken from [1] and [7].
- $G(A)$ is triangle-free [1].
- **Cycle lemma** [1, Thm 4(b)]: if $A\setminus\{u\}$, $A\setminus\{v\}$ and $A\setminus\{w\}$ are all regular for a sextuple $A$, then $S(Z)=0$ for $Z=A\setminus\{u,v,w\}$.

For a septuple, a vertex $x$ of degree four with neighbours $c_1,\dots,c_4$ and non-neighbours $a,b$ forces $S(a,b,c_\ell)=0$ for each $\ell$. So the $c_\ell$ are the four roots of $S(a,b,\cdot)$, and in particular $c_4=2(a+b)-c_1-c_2-c_3$.

**Degree lemma** [7]: no vertex has degree five or more. With the root sums, five regular fifths of a sextuple would force two of its entries to coincide.

### 2.3 At most one vertex of degree four [proved]

Suppose $x\neq x'$ both have degree four. They are non-adjacent, and $|N(x)\cap N(x')|\ge 3$.

- **Case $N(x)=N(x')$.** The common non-neighbour $w$ makes $S(x,w,\cdot)$ and $S(x',w,\cdot)$ share four roots. Comparing root sums gives $x=x'$.
- **Case $|N(x)\cap N(x')|=3$.** The two monic quartics share a cubic factor, which gives five polynomial equations. The leading equation gives $c_4=w-\tfrac23(x-x')$. After cancelling $x-x'$ and saturating by $x x' w (x-x')$, the Gröbner basis is $\{1\}$ over $\mathbb Q$, and also on the branch $x'=\pm i$ over $\mathbb Q(i)$. The computation is exact (sympy; `checks/deg4_pair*.py`).

This excludes the seven census graphs with at least two degree-four vertices (rows 73, 74, 91, 92, 102, 106, 107 of the census `rank/shapes.txt` of the 107 triangle-free graphs on seven vertices). Graphs with a vertex of degree at least five (rows 17, 31, 32, 52, 72, 90, 101) are excluded by the degree lemma. The lemma uses only five of the regular fifths, so it also excludes degree six.

### 2.4 The system for one graph

Label the degree-four vertex 0 = $x$, its non-neighbours 1 = $a$ and 2 = $b$, and its neighbours 3–6 = $c_1,\dots,c_4$. For a census graph with unique degree-four vertex, the system in the variables $x,a,b,c_1,c_2,c_3$ consists of:
- (R) $S(a,b,c_1)=0$, together with the successive divided differences that make $c_2$ and $c_3$ further roots;
- (C) $S(Z_v)=0$ for each vertex $v\ne x$ and each 3-subset $T\subseteq N(v)$, with $Z_v$ the complement of $\{v\}\cup T$;
- (P) $P_5(A\setminus\{u,v\})=0$ for every edge $uv$ not through $x$;
- (H) $t_0H-1=0$, with $H$ the product of the seven entries and either all 21 differences ("full") or a chosen subset of them (Section 3.2).

Every equation in (R), (C) and (P) is a necessary condition for a septuple whose graph contains the given edges. Two remarks on the scope of the system.
- The system imposes the edges of the row, not the absence of other edges. A configuration found on row $\Gamma$ may have a larger graph $\Gamma'\supseteq\Gamma$. Section 5.3 computes the exact graph on each orbit.
- No square condition is imposed: the system involves only the entries, not the square roots of the 21 values. Squares enter in Section 4.

## 3. The census

### 3.1 The 23 rows

In the labelling of 2.4, $x$ is joined to $c_1,\dots,c_4$ in every row. The further edges are:

| rows | extra edges |
|---|---|
| 9 | none |
| 18, 19 | $ac_1$; $ab$ |
| 33, 34, 35, 39 | $\{ac_1,ac_2\}$; $\{ac_1,bc_1\}$; $\{ac_1,ab\}$; $\{bc_1,ac_2\}$ |
| 53, 54, 55, 56, 57 | $\{ac_1,ac_2,ac_3\}$; $\{ac_1,ac_2,ab\}$; $\{ac_1,bc_1,ac_2\}$; $\{ab,ac_2,bc_1\}$; $\{ac_2,ac_3,bc_1\}$ |
| 75–79, 84 | 4 extra edges |
| 93–95 | 5 extra edges |
| 103, 104 | 6 extra edges |

The machine-readable list is `checks/deg4_rows.txt`. Since $S(a,b,\cdot)$ is symmetric in its roots and the system is symmetric under $a\leftrightarrow b$, each row is one representative of its orbit under $S_2\times S_4$.

### 3.2 Method

- Each row is first solved modulo $p=536870909$ (Singular `std`, then saturation).
  - Saturating by the seven entries alone usually leaves components of dimension 1 or 2. On these, two entries coincide.
  - `checks/deg4_hit.py` then searches for a smallest set $D$ of differences such that saturating by the entries and $\prod D$ gives a zero-dimensional or empty ideal.
- The row is then solved in characteristic zero by msolve with $H$ = entries $\cdot\prod D$, or $H$ = entries $\cdot$ all 21 differences for the empty rows.
- For finite rows, msolve returns a rational parametrisation.
  - `checks/deg4_Qpar.py` factors the eliminant over $\mathbb Q$.
  - It verifies every equation exactly in each field $K=\mathbb Q[T]/(g)$.
  - It checks that one input coordinate generates $K$, so each factor contributes $\deg g$ distinct points.
- A point found with a partial $D$ is kept only if all seven entries are distinct and nonzero in $K$. This is checked exactly (`checks/deg4_field.py`).

### 3.3 Results

**Ten empty rows** [proved mod msolve]: 75, 76, 77, 78, 84, 93, 94, 95, 103, 104.
- msolve over $\mathbb Q$ with the full $H$ returns $[-1]$ in 11–33 s per row.
- The count over $\mathbb F_p$ is zero for $p=101,211,401,809,1201$ (`checks/deg4_fp.c`), and the Gröbner basis modulo 536870909 is $\{1\}$.
- An exact `std` over $\mathbb Q$ did not finish: row 78 ran past 3600 s. The label therefore stays "mod msolve".

**Finite rows**: 53, 54, 55, 56, 57, 79 [proved mod msolve for completeness; every listed point is verified exactly].

| row | $D$ | points over $\overline{\mathbb Q}$ | Galois orbits | fields: signature; $\lvert\mathrm{disc}\rvert$ |
|---|---|---|---|---|
| 79 | all 21 | 8 | 1 of degree 8 | (4,2); $2^{12}3^65^4$ |
| 53 | $x-a$ | 48 | 6 of degree 8 | all (0,4); $2^{16}3^6$ |
| 55 | $x-a,x-b,a-b$ | 56 | 7 of degree 8 | (4,2) $2^83^417^4$ ×2; (0,4) $2^83^613^4$ ×2; (4,2) $2^{12}3^65^4$ ×2; (4,2) $2^{12}3^423^4$ |
| 54 | $x-a,x-b,a-c_3$ | 80 | 20 of degree 4 | (2,1) $2^33^219^2$ ×4; (2,1) $2^63^27$ ×8; (0,2) $2^33^35^2$ ×8 |
| 57 | $x-a,x-b,a-b,a-c_1$ | 84 | 9 of degree 8, 1 of degree 12 | degree 8 as in rows 79, 53, 55; degree 12: (0,6) |
| 56 | $x-a,x-b,x-c_1,\dots,x-c_4,a-b,a-c_1$ | 64 | 16 of degree 4 | (4,0) $3^27^223^2$ ×4; (2,1) $3^343^2$ ×8; (2,1) $3^211^3$ ×4 |

- For rows 53, 54, 55, 56 and 57, the vector-space dimension of the radical modulo $p$ equals the characteristic-zero degree. This is an independent check that the list is complete.
- No orbit is rational. The smallest degree is 4 (rows 54 and 56).
- The degree-8 fields all have subfield degrees $[1,2,2,2,4,4,4,8]$ [heuristic remark: consistent with a 2-group Galois closure; not used].

**Seven remaining rows**: 9, 18, 19 and 33, 34, 35, 39.
- The expected dimensions are 3, 2, 2 for rows 9, 18, 19 and 1 for rows 33–39: six unknowns, three root conditions, and one $P_5$ per extra edge. No vertex other than $x$ has degree three, so (C) is empty.
- Row 9 has dimension 3 (Krull dimension modulo 536870909, `checks/deg4_modp.py`).
- For the other rows the dimension is supported by point counts [heuristic, Lang–Weil]. The number of $\mathbb F_p$-points with distinct nonzero entries, divided by $p^2$, is 0.86, 0.93, 0.94 for row 18 at $p=401,809,1201$ (twice that for row 19). For rows 33–39 the count grows linearly in $p$ (944–2844 points at $p=401$, 2584–8784 at $p=1201$; `checks/deg4_fp.log`). Fixing $x=3$ on rows 33 and 39 gives a nonempty finite set modulo 65521 (96 and 104 points).

**Consequence** [proved mod msolve]. Let $A$ be a rational septuple whose graph $G(A)$ has a vertex of degree four. Then $G(A)$ is isomorphic to one of rows 9, 18, 19, 33, 34, 35, 39.

Proof: by Section 2.3 and the degree lemma, $G(A)$ is one of the 23 rows. The system of each other row has no rational solution with distinct nonzero entries. That system only imposes edges, so this also excludes every graph that contains such a row.

## 4. Square-class quotients

### 4.1 The relations

This is the computation of [3, Lemma B], applied to the degree-four rows. Let $W$ be a regular quintuple and let $\{u,v\}\subset W$ be a pair with complement $T=\{p,q,r\}$. By the form of $P_5$ in 2.1 and its symmetry, $(pq+1)(pr+1)(qr+1)(uv+1)$ is a square. If all 21 values $w_{ij}=e_ie_j+1$ are nonzero, this gives the relation
$$[pq]+[pr]+[qr]+[uv]=0\quad\text{in } K^*/K^{*2}$$
for any field $K$ containing the entries. Each edge of $G(A)$ gives ten such relations. Let $V_\Gamma$ be the quotient of $\mathbb F_2^{21}$ by the relations of all edges of $\Gamma$, and let $r=\dim V_\Gamma$.

- [proved] Suppose the graph of $A$ contains $\Gamma$ and the 21 values are nonzero. Then the map $\mathbb F_2^{21}\to K^*/K^{*2}$ factors through $V_\Gamma$. So the 21 classes are determined by $r$ values whose images form a basis of $V_\Gamma$. (For every row below, each pair maps to a basis vector, and the classes in the table are the fibres.) $A$ is a Diophantine septuple over $K$ if and only if these $r$ values are squares in $K$.
- The rank $s$ of the image in $K^*/K^{*2}$ satisfies $s\le r$. Further relations can only lower $s$.
- The rank $r$ is a property of $\Gamma$, computed by Gaussian elimination (`checks/deg4_sqclass.py`).

### 4.2 Ranks

| row | $r$ | classes |
|---|---|---|
| 9 | 7 | the 15 pairs inside $\{a,b,c_1,\dots,c_4\}$; $xa$; $xb$; $xc_1$; $xc_2$; $xc_3$; $xc_4$ |
| 18 | 3 | 19 pairs; $xa$; $xc_1$ |
| 19 | 3 | 19 pairs; $xa$; $xb$ |
| 33, 35 | 2 | 20 pairs; $xa$ |
| 34 | 2 | 20 pairs; $xc_1$ |
| 39 | 1 | all 21 |
| 53, 54 | 2 | 20 pairs; $xa$ |
| 55, 56, 57, 79 | 1 | all 21 |

In row 9 the six values $xa,\dots,xc_4$ are free. The degree-four vertex alone ties the 15 values of the sextuple $A\setminus\{x\}$ into one class. Each extra edge then removes free classes.

Cross-checks.
- Over $\mathbb F_p$, I enumerated all points of rows 18–39 with distinct nonzero entries and nonzero values (`checks/deg4_fp.c`), and took the Legendre vectors of the 21 values (`checks/deg4_legendre.py`) [heuristic as evidence for components not seen mod $p$].
  - At $p=101$ there are 4488, 9792, 128, 216, 492 and 64 points on rows 18, 19, 33, 34, 35 and 39.
  - The $\mathbb F_2$-span of the vectors has rank 3, 3, 2, 2, 2, 1, and the partition of the 21 columns into equal columns is exactly the class partition of the table.
  - The same ranks and partitions hold at $p=401$ (between 944 and 274464 points per row) and $p=809$ (between 2016 and 1211328 points).
  - Row 39 has exactly two distinct Legendre vectors at each prime: all $+1$, and all $-1$. So the curve has $\mathbb F_p$-points on which every value is a square, and points on which none is.
- On every finite orbit, the classes hold exactly in $K$, and the rank $s$ of the representatives in $K^*/K^{*2}$ equals $r$ (`checks/deg4_sqrank.py`). The one exception is the degree-12 orbit of row 57: there $xa+1=0$, so the relations do not apply to that value.

### 4.3 Row 39

**Corollary** [proved mod msolve for the reduction of Section 3; the rest is proved]. Let $C_{39}$ be the curve of row 39 over $\mathbb Q$. A rational point of $C_{39}$ with distinct nonzero entries, all 21 values nonzero, and $xa+1\in\mathbb Q^{*2}$ is a rational Diophantine septuple. Conversely, a rational septuple with $G(A)\cong$ row 39 gives such a point.

So row 39 reduces to one square condition on a curve. Its rational points, genus and components over $\mathbb Q$ are not determined here (Section 6). For rows 33, 34 and 35 the condition is two squares on a curve.

## 5. Septuples over number fields

### 5.1 Construction

For a finite orbit with field $K$ and $s$ independent classes, let $L=K(\sqrt{w_1},\dots,\sqrt{w_s})$ for representatives $w_i$. By 4.1 every $w_{ij}$ is a square in $L$, so the orbit is a Diophantine septuple over $L$, of degree $[K:\mathbb Q]\,2^s$.

`checks/deg4_septL.py` builds $L$ as an absolute field and checks directly, without using 4.1, that all 21 values are squares in $L$ and that the seven entries are distinct and nonzero. This check is exact and does not depend on msolve: each point is verified by substitution.

### 5.2 Results [proved]

| row | orbit | $[K:\mathbb Q]$ | $s$ | $[L:\mathbb Q]$ | signature of $L$ |
|---|---|---|---|---|---|
| 79 | the unique orbit | 8 | 1 ($xa$) | 16 | (8,4) |
| 54 | factor 0 | 4 | 2 ($xa,xb$) | 16 | (8,4) |
| 54 | factor 2 | 4 | 2 | 16 | (8,4) |
| 54 | factor 6 | 4 | 2 | 16 | (0,8) |
| 56 | factor 0 | 4 | 1 ($xa$) | 8 | (8,0) |
| 56 | factor 2 | 4 | 1 | 8 | (4,2) |
| 56 | factor 6 | 4 | 1 | 8 | (4,2) |

The three row-54 factors, and the three row-56 factors, are one from each discriminant class in 3.3. The other row-56 factors also have $s=1$ (`deg4_sqrank.py`), so they too give septuples over octic fields. The other factors of row 54, and the degree-8 orbits of rows 55 and 57, give septuples over degree-16 fields by 4.1, since $s=1$ for those orbits of rows 55 and 57. These were not rebuilt by `deg4_septL.py`. Row 53 gives degree 32.

Row 79 in closed form. Take $K=\mathbb Q(T)$ with
$$T^8-\tfrac{219}{10}T^6+\tfrac{27081}{400}T^4-\tfrac{9747}{40}T^2+\tfrac{81}{16}=0,$$
and $a=T$. The other coordinates are odd polynomials in $T$ of degree at most 7 (supplement, `deg4_Qpar.jsonl`). So $T\mapsto -T$ sends $A$ to $-A$, and the 21 values lie in the quartic subfield $\mathbb Q(T^2)$. That subfield is the $D_4$ field $u^4-2u^3-3u^2+4u-2$, of discriminant $-2^6\cdot3^3\cdot5$. A polredabs model of $L$ is in the supplement.

Row 56, factor 0, in closed form. Let $K=\mathbb Q(u)$ with $u^4-u^3-52u^2-155u-125=0$. This field is $\mathbb Q(\sqrt{21},\sqrt{69})$, totally real with discriminant $3^2 7^2 23^2$. Then
$$\begin{aligned}
x&=\tfrac{1}{425}(-147u^3+882u^2+4704u)-\tfrac{1393}{85}, & a&=\tfrac{5793}{8500}u^3-\tfrac{3547}{2125}u^2-\tfrac{69379}{2125}u-\tfrac{18971}{340},\\
b&=-\tfrac{381}{425}u^3+\tfrac{503}{170}u^2+\tfrac{16799}{425}u+\tfrac{8509}{170}, & c_1&=\tfrac{489}{500}u^3-\tfrac{2017}{875}u^2-\tfrac{41919}{875}u-\tfrac{12013}{140},\\
c_2&=-\tfrac{32}{25}u^3+\tfrac{1441}{350}u^2+\tfrac{403}{7}u+\tfrac{985}{14}, & c_3&=-\tfrac{956}{14875}u^3+\tfrac{10917}{29750}u^2+\tfrac{33547}{14875}u+\tfrac{411}{170},\\
c_4&=-\tfrac{757}{11900}u^3+\tfrac{1191}{2975}u^2+\tfrac{1093}{595}u+\tfrac{105}{68}.
\end{aligned}$$
`checks/deg4_row56_example.py` checks the following in PARI, independently of Sage and msolve:
- $c_1,\dots,c_4$ are the roots of $S(a,b,\cdot)$;
- the regular fifths are exactly the seven edges of row 56;
- the entries are distinct and nonzero;
- $xa+1$ is not a square in $K$, but each of the 21 values divided by $xa+1$ is.

So the seven entries form a Diophantine septuple over the totally real octic field $K(\sqrt{xa+1})$, with polredabs model $s^8-112s^6+2886s^4-22225s^2+15625$.

### 5.3 The exact graph

On each orbit I computed $G(A)$ by testing $P_5=0$ in $K$ on all 21 five-subsets. Since $P_5$ is symmetric, one ordering per subset suffices. Regularity is an algebraic condition, so the graph over $L$ is the same.
- Rows 79, 56, 54 and 53: $G(A)$ is exactly the row on every orbit (`checks/deg4_graph.py` for row 56).
- Row 55: some orbits gain one edge ($bc_3$ or $bc_4$), and they then have the row-79 graph. These are the orbits whose field is the row-79 field.
- Row 57: the orbits have 7 or 8 edges.

So the septuples of 5.2 have a unique vertex of degree four. Their graphs are rows 79, 56 and 54. Earlier septuples over number fields have graph a 4-cycle [10] or a 7-cycle [3]. Their fields are $\mathbb Q(i)$ and fields of degree 8 and 16.

## 6. Limitations and open rows

- No rational septuple is found and none is excluded. Every negative statement here is about one row's system, i.e. about septuples whose graph contains that row.
- Completeness of the finite lists and emptiness of the ten zero rows rest on msolve's multimodular characteristic-zero computation. Each listed point is verified exactly. An exact Gröbner basis over $\mathbb Q$ was infeasible (row 78: more than 3600 s).
- The seven remaining rows are not resolved.
  - Row 39 is a curve on which one square condition is equivalent to a rational septuple (4.3). Its genus, its components over $\mathbb Q$ and its local solubility are the next step.
  - Rows 33, 34, 35 are curves with two square conditions.
  - Rows 9, 18, 19 have dimension 3, 2, 2 and $r=7,3,3$.
- All seven remaining rows lie over the same base. Deleting $x$ from a septuple of any of them leaves a sextuple $\{a,b,c_1,\dots,c_4\}$ with $ab+1=\square$ and $S(a,b,\cdot)$ split over $\mathbb Q$, i.e. a rational point of the genus-5 fibres $X_t$ of [8]. I searched these points through the elliptic fibration of [8]: 474 fibres (one per value of $\kappa^2$, base height at most 45), all combinations $\sum n_iP_i+T$ of the ellrank generators and torsion with $|n_i|\le 8$ (or $\le 4$ when there are at least two generators), 645,792 points (`checks/x4_fibsearch.py`, log in the supplement). Every point with $\Delta$ a square, 3248 of them, has $ab=-3/4$ and the completion $c=0$. That is the zero boundary, which is not admissible. No rational sextuple with four regular fifths is known. [search bound only]
- The Legendre comparisons in 4.2 are evidence, not proof. The combinatorial rank is [proved] as an upper bound only. It is attained on every finite orbit, but on the positive-dimensional rows an extra relation on some component over $\mathbb Q$ is not excluded.
- The Galois-structure remark in 3.3 is [heuristic].

## 7. Reproduction

All scripts are in the supplement. Run them from its root, with Python 3, passagemath (Singular, number fields), cypari2 and msolve 0.10.1.

- `checks/deg4_rows.txt`: the 23 rows. `checks/deg4_pair*.py`: Section 2.3.
- `checks/deg4_Q.py ROW full|D:...`: msolve systems; `checks/deg4_hit.py`: hitting sets $D$ mod $p$.
- `checks/deg4_Qpar.py`, `checks/deg4_field.py`: exact verification of each orbit; output `deg4_Qpar.jsonl`, `deg4_field.jsonl`.
- `checks/deg4_fp.c`: $\mathbb F_p$ enumeration. `checks/deg4_legendre.py P ROW...`: Legendre vectors.
- `checks/deg4_sqclass.py`: $r$ and the class partition. `checks/deg4_sqrank.py ROW`: classes and rank in $K$.
- `checks/deg4_septL.py ROW FACTOR`: the septuples over $L$; output `deg4_septL.jsonl` (with defining polynomials of $L$).
- `checks/deg4_slice.py`: one-coordinate slices of the curve rows.
- `checks/deg4_graph.py ROW`: the exact graph $G(A)$ on each orbit and its census row; output `deg4_graph.jsonl`.
- `checks/deg4_row56_example.py`: the PARI check of the row-56 closed form in 5.2.
- `checks/x4_fibsearch.py 45 8`: the fibration search of Section 6; output `x4_fib_H45N8.jsonl` and `.log`.
- `checks/ms/*.ms.out`: the msolve outputs of Section 3 (SHA-256 sums of the inputs in `checks/ms/INPUT_SHA256.txt`).

## References

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2. A. Dujella, *On Diophantine quintuples*, Acta Arith. 81 (1997) 69–79. https://doi.org/10.4064/aa-81-1-69-79
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11. J. Berthomieu, C. Eder, M. Safey El Din, *msolve: a library for solving polynomial systems*, ISSAC 2021, 51–58. https://doi.org/10.1145/3452143.3465545
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