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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 3450.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3450.j1 | 3450j3 | \([1, 0, 1, -34560276, -78204168302]\) | \(292169767125103365085489/72534787200\) | \(1133356050000000\) | \([2]\) | \(172032\) | \(2.7043\) | |
| 3450.j2 | 3450j4 | \([1, 0, 1, -2528276, -777224302]\) | \(114387056741228939569/49503729150000000\) | \(773495767968750000000\) | \([2]\) | \(172032\) | \(2.7043\) | |
| 3450.j3 | 3450j2 | \([1, 0, 1, -2160276, -1221768302]\) | \(71356102305927901489/35540674560000\) | \(555323040000000000\) | \([2, 2]\) | \(86016\) | \(2.3578\) | |
| 3450.j4 | 3450j1 | \([1, 0, 1, -112276, -25736302]\) | \(-10017490085065009/12502381363200\) | \(-195349708800000000\) | \([2]\) | \(43008\) | \(2.0112\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3450.j have rank \(1\).
Complex multiplication
The elliptic curves in class 3450.j do not have complex multiplication.Modular form 3450.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
