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SageMath
E = EllipticCurve("eq1")
E.isogeny_class()
Elliptic curves in class 235950eq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 235950.eq7 | 235950eq1 | \([1, 1, 1, -34545563, 74743938281]\) | \(164711681450297281/8097103872000\) | \(224133022384128000000000\) | \([4]\) | \(39813120\) | \(3.2399\) | \(\Gamma_0(N)\)-optimal |
| 235950.eq6 | 235950eq2 | \([1, 1, 1, -96497563, -268346237719]\) | \(3590017885052913601/954068544000000\) | \(26409228498081000000000000\) | \([2, 2]\) | \(79626240\) | \(3.5864\) | |
| 235950.eq3 | 235950eq3 | \([1, 1, 1, -2764305563, 55939456578281]\) | \(84392862605474684114881/11228954880\) | \(310824664627620000000\) | \([4]\) | \(119439360\) | \(3.7892\) | |
| 235950.eq8 | 235950eq4 | \([1, 1, 1, 243270437, -1735464461719]\) | \(57519563401957999679/80296734375000000\) | \(-2222665047595458984375000000\) | \([2]\) | \(159252480\) | \(3.9330\) | |
| 235950.eq5 | 235950eq5 | \([1, 1, 1, -1427497563, -20757760237719]\) | \(11621808143080380273601/1335706803288000\) | \(36973220002182696375000000\) | \([2]\) | \(159252480\) | \(3.9330\) | |
| 235950.eq2 | 235950eq6 | \([1, 1, 1, -2764547563, 55929172062281]\) | \(84415028961834287121601/30783551683856400\) | \(852108431321942622506250000\) | \([2, 2]\) | \(238878720\) | \(4.1357\) | |
| 235950.eq4 | 235950eq7 | \([1, 1, 1, -2365792063, 72629849913281]\) | \(-52902632853833942200321/51713453577420277500\) | \(-1431461523954191316065273437500\) | \([2]\) | \(477757440\) | \(4.4823\) | |
| 235950.eq1 | 235950eq8 | \([1, 1, 1, -3167175063, 38570290027281]\) | \(126929854754212758768001/50235797102795981820\) | \(1390559046112911755453453437500\) | \([2]\) | \(477757440\) | \(4.4823\) |
Rank
sage: E.rank()
The elliptic curves in class 235950eq have rank \(0\).
Complex multiplication
The elliptic curves in class 235950eq do not have complex multiplication.Modular form 235950.2.a.eq
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
