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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 10350.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10350.f1 | 10350l4 | \([1, -1, 0, -173367, -4703459]\) | \(50591419971625/28422890688\) | \(323754489243000000\) | \([2]\) | \(110592\) | \(2.0500\) | |
10350.f2 | 10350l2 | \([1, -1, 0, -129492, -17903084]\) | \(21081759765625/57132\) | \(650769187500\) | \([2]\) | \(36864\) | \(1.5007\) | |
10350.f3 | 10350l1 | \([1, -1, 0, -7992, -285584]\) | \(-4956477625/268272\) | \(-3055785750000\) | \([2]\) | \(18432\) | \(1.1542\) | \(\Gamma_0(N)\)-optimal |
10350.f4 | 10350l3 | \([1, -1, 0, 42633, -599459]\) | \(752329532375/448524288\) | \(-5108971968000000\) | \([2]\) | \(55296\) | \(1.7035\) |
Rank
sage: E.rank()
The elliptic curves in class 10350.f have rank \(0\).
Complex multiplication
The elliptic curves in class 10350.f do not have complex multiplication.Modular form 10350.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.