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SageMath
E = EllipticCurve("fi1")
E.isogeny_class()
Elliptic curves in class 85680fi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85680.dp4 | 85680fi1 | \([0, 0, 0, 2013, 58466]\) | \(302111711/669375\) | \(-1998743040000\) | \([2]\) | \(131072\) | \(1.0445\) | \(\Gamma_0(N)\)-optimal |
| 85680.dp3 | 85680fi2 | \([0, 0, 0, -15987, 638066]\) | \(151334226289/28676025\) | \(85626151833600\) | \([2, 2]\) | \(262144\) | \(1.3910\) | |
| 85680.dp2 | 85680fi3 | \([0, 0, 0, -77187, -7672894]\) | \(17032120495489/1339001685\) | \(3998237607383040\) | \([2]\) | \(524288\) | \(1.7376\) | |
| 85680.dp1 | 85680fi4 | \([0, 0, 0, -242787, 46043426]\) | \(530044731605089/26309115\) | \(78558596444160\) | \([2]\) | \(524288\) | \(1.7376\) |
Rank
sage: E.rank()
The elliptic curves in class 85680fi have rank \(1\).
Complex multiplication
The elliptic curves in class 85680fi do not have complex multiplication.Modular form 85680.2.a.fi
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
