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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 257600.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 257600.z1 | 257600z4 | \([0, 1, 0, -266929633, -1678668331137]\) | \(513516182162686336369/1944885031250\) | \(7966249088000000000000\) | \([2]\) | \(69009408\) | \(3.4171\) | |
| 257600.z2 | 257600z3 | \([0, 1, 0, -16929633, -25418331137]\) | \(131010595463836369/7704101562500\) | \(31556000000000000000000\) | \([2]\) | \(34504704\) | \(3.0705\) | |
| 257600.z3 | 257600z2 | \([0, 1, 0, -4545633, -401019137]\) | \(2535986675931409/1450751712200\) | \(5942279013171200000000\) | \([2]\) | \(23003136\) | \(2.8678\) | |
| 257600.z4 | 257600z1 | \([0, 1, 0, -2945633, 1936580863]\) | \(690080604747409/3406760000\) | \(13954088960000000000\) | \([2]\) | \(11501568\) | \(2.5212\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 257600.z have rank \(1\).
Complex multiplication
The elliptic curves in class 257600.z do not have complex multiplication.Modular form 257600.2.a.z
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.
