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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 325703.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 325703.e1 | 325703e4 | \([1, -1, 1, -1751539, -891590500]\) | \(209267191953/55223\) | \(156820113711182663\) | \([2]\) | \(4915200\) | \(2.2839\) | |
| 325703.e2 | 325703e2 | \([1, -1, 1, -123024, -10238182]\) | \(72511713/25921\) | \(73609441129738801\) | \([2, 2]\) | \(2457600\) | \(1.9374\) | |
| 325703.e3 | 325703e1 | \([1, -1, 1, -52219, 4489258]\) | \(5545233/161\) | \(457201497700241\) | \([2]\) | \(1228800\) | \(1.5908\) | \(\Gamma_0(N)\)-optimal |
| 325703.e4 | 325703e3 | \([1, -1, 1, 372611, -72291684]\) | \(2014698447/1958887\) | \(-5562770622518832247\) | \([2]\) | \(4915200\) | \(2.2839\) |
Rank
sage: E.rank()
The elliptic curves in class 325703.e have rank \(1\).
Complex multiplication
The elliptic curves in class 325703.e do not have complex multiplication.Modular form 325703.2.a.e
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.
