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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 11310e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
11310.d2 | 11310e1 | \([1, 0, 1, -4170829, 3278200952]\) | \(8023996232564328604273609/2693913066000\) | \(2693913066000\) | \([6]\) | \(214272\) | \(2.1809\) | \(\Gamma_0(N)\)-optimal |
11310.d3 | 11310e2 | \([1, 0, 1, -4170249, 3279158416]\) | \(-8020649220830773808798089/4649360115706312500\) | \(-4649360115706312500\) | \([6]\) | \(428544\) | \(2.5275\) | |
11310.d1 | 11310e3 | \([1, 0, 1, -4240444, 3163087226]\) | \(8432523527010257294720569/556754628456000000000\) | \(556754628456000000000\) | \([2]\) | \(642816\) | \(2.7302\) | |
11310.d4 | 11310e4 | \([1, 0, 1, 3564036, 13489975162]\) | \(5006683449688877689783751/81509038330078125000000\) | \(-81509038330078125000000\) | \([2]\) | \(1285632\) | \(3.0768\) |
Rank
sage: E.rank()
The elliptic curves in class 11310e have rank \(1\).
Complex multiplication
The elliptic curves in class 11310e do not have complex multiplication.Modular form 11310.2.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 Cremona 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 Cremona labels.