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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 57960k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 57960.b4 | 57960k1 | \([0, 0, 0, -1623, -437078]\) | \(-2533446736/440749575\) | \(-82254448684800\) | \([2]\) | \(163840\) | \(1.3494\) | \(\Gamma_0(N)\)-optimal |
| 57960.b3 | 57960k2 | \([0, 0, 0, -96843, -11501642]\) | \(134555337776164/1312250625\) | \(979589842560000\) | \([2, 2]\) | \(327680\) | \(1.6960\) | |
| 57960.b2 | 57960k3 | \([0, 0, 0, -171363, 8633662]\) | \(372749784765122/194143359375\) | \(289854482400000000\) | \([2]\) | \(655360\) | \(2.0426\) | |
| 57960.b1 | 57960k4 | \([0, 0, 0, -1545843, -739769042]\) | \(273629163383866082/26408025\) | \(39426970060800\) | \([2]\) | \(655360\) | \(2.0426\) |
Rank
sage: E.rank()
The elliptic curves in class 57960k have rank \(0\).
Complex multiplication
The elliptic curves in class 57960k do not have complex multiplication.Modular form 57960.2.a.k
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.
