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SageMath
E = EllipticCurve("eb1")
E.isogeny_class()
Elliptic curves in class 87360.eb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 87360.eb1 | 87360fu8 | \([0, 1, 0, -2572189121, 50210557479999]\) | \(7179471593960193209684686321/49441793310\) | \(12960869465456640\) | \([2]\) | \(21233664\) | \(3.6275\) | |
| 87360.eb2 | 87360fu6 | \([0, 1, 0, -160761921, 784498732479]\) | \(1752803993935029634719121/4599740941532100\) | \(1205794489376990822400\) | \([2, 2]\) | \(10616832\) | \(3.2809\) | |
| 87360.eb3 | 87360fu7 | \([0, 1, 0, -158786241, 804722188095]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-23567135137169030184960000\) | \([2]\) | \(21233664\) | \(3.6275\) | |
| 87360.eb4 | 87360fu5 | \([0, 1, 0, -31769921, 68800473279]\) | \(13527956825588849127121/25701087819771000\) | \(6737385965426049024000\) | \([2]\) | \(7077888\) | \(3.0782\) | |
| 87360.eb5 | 87360fu3 | \([0, 1, 0, -10171201, 11938220735]\) | \(443915739051786565201/21894701746029840\) | \(5739564694511246376960\) | \([2]\) | \(5308416\) | \(2.9344\) | |
| 87360.eb6 | 87360fu2 | \([0, 1, 0, -2649921, 292761279]\) | \(7850236389974007121/4400862921000000\) | \(1153659809562624000000\) | \([2, 2]\) | \(3538944\) | \(2.7316\) | |
| 87360.eb7 | 87360fu1 | \([0, 1, 0, -1646401, -809304385]\) | \(1882742462388824401/11650189824000\) | \(3054027361222656000\) | \([2]\) | \(1769472\) | \(2.3850\) | \(\Gamma_0(N)\)-optimal |
| 87360.eb8 | 87360fu4 | \([0, 1, 0, 10413759, 2333308095]\) | \(476437916651992691759/284661685546875000\) | \(-74622352896000000000000\) | \([2]\) | \(7077888\) | \(3.0782\) |
Rank
sage: E.rank()
The elliptic curves in class 87360.eb have rank \(1\).
Complex multiplication
The elliptic curves in class 87360.eb do not have complex multiplication.Modular form 87360.2.a.eb
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
