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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 12705.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12705.f1 | 12705o3 | \([1, 0, 0, -69577120, -223387596475]\) | \(21026497979043461623321/161783881875\) | \(286610015558356875\) | \([2]\) | \(921600\) | \(2.9422\) | |
| 12705.f2 | 12705o2 | \([1, 0, 0, -4351465, -3485823208]\) | \(5143681768032498601/14238434358225\) | \(25224255010091439225\) | \([2, 2]\) | \(460800\) | \(2.5957\) | |
| 12705.f3 | 12705o4 | \([1, 0, 0, -2636290, -6261319393]\) | \(-1143792273008057401/8897444448004035\) | \(-15762365583750476248635\) | \([2]\) | \(921600\) | \(2.9422\) | |
| 12705.f4 | 12705o1 | \([1, 0, 0, -382060, -6242785]\) | \(3481467828171481/2005331497785\) | \(3552567073547492385\) | \([4]\) | \(230400\) | \(2.2491\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12705.f have rank \(0\).
Complex multiplication
The elliptic curves in class 12705.f do not have complex multiplication.Modular form 12705.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
