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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 11970.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11970.bm1 | 11970bg4 | \([1, -1, 1, -7898123, -8541482669]\) | \(2768241956450868452043/2058557375000\) | \(40518584812125000\) | \([2]\) | \(373248\) | \(2.4967\) | |
| 11970.bm2 | 11970bg3 | \([1, -1, 1, -490403, -135202013]\) | \(-662660286993086283/18441985352000\) | \(-362993597683416000\) | \([2]\) | \(186624\) | \(2.1501\) | |
| 11970.bm3 | 11970bg2 | \([1, -1, 1, -118883, -6177373]\) | \(6882017790203934867/3366201047283200\) | \(90887428276646400\) | \([6]\) | \(124416\) | \(1.9474\) | |
| 11970.bm4 | 11970bg1 | \([1, -1, 1, 27037, -749149]\) | \(80956273702840173/55667967918080\) | \(-1503035133788160\) | \([6]\) | \(62208\) | \(1.6008\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 11970.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 11970.bm do not have complex multiplication.Modular form 11970.2.a.bm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
