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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 141570.dp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 141570.dp1 | 141570ca7 | \([1, -1, 1, -1140183023, -8331182645893]\) | \(126929854754212758768001/50235797102795981820\) | \(64877922855444010862436323580\) | \([2]\) | \(159252480\) | \(4.2269\) | |
| 141570.dp2 | 141570ca6 | \([1, -1, 1, -995237123, -12080701165453]\) | \(84415028961834287121601/30783551683856400\) | \(39755970971756554995651600\) | \([2, 2]\) | \(79626240\) | \(3.8803\) | |
| 141570.dp3 | 141570ca3 | \([1, -1, 1, -995150003, -12082922620909]\) | \(84392862605474684114881/11228954880\) | \(14501835552866238720\) | \([2]\) | \(39813120\) | \(3.5338\) | |
| 141570.dp4 | 141570ca8 | \([1, -1, 1, -851685143, -15688047581269]\) | \(-52902632853833942200321/51713453577420277500\) | \(-66786268861606750042341397500\) | \([2]\) | \(159252480\) | \(4.2269\) | |
| 141570.dp5 | 141570ca4 | \([1, -1, 1, -513899123, 4483676211347]\) | \(11621808143080380273601/1335706803288000\) | \(1725022552421835882072000\) | \([2]\) | \(53084160\) | \(3.6776\) | |
| 141570.dp6 | 141570ca2 | \([1, -1, 1, -34739123, 57962787347]\) | \(3590017885052913601/954068544000000\) | \(1232148964806467136000000\) | \([2, 2]\) | \(26542080\) | \(3.3310\) | |
| 141570.dp7 | 141570ca1 | \([1, -1, 1, -12436403, -16144690669]\) | \(164711681450297281/8097103872000\) | \(10457150292353875968000\) | \([2]\) | \(13271040\) | \(2.9845\) | \(\Gamma_0(N)\)-optimal |
| 141570.dp8 | 141570ca5 | \([1, -1, 1, 87577357, 374860323731]\) | \(57519563401957999679/80296734375000000\) | \(-103700660460613734375000000\) | \([2]\) | \(53084160\) | \(3.6776\) |
Rank
sage: E.rank()
The elliptic curves in class 141570.dp have rank \(0\).
Complex multiplication
The elliptic curves in class 141570.dp do not have complex multiplication.Modular form 141570.2.a.dp
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
