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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 30345.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
30345.e1 | 30345f2 | \([1, 1, 1, -2959366, -1875442666]\) | \(24170156844497/1191196125\) | \(141261418955204974125\) | \([2]\) | \(940032\) | \(2.6257\) | |
30345.e2 | 30345f1 | \([1, 1, 1, 111259, -114132166]\) | \(1284365503/48234375\) | \(-5720012105409984375\) | \([2]\) | \(470016\) | \(2.2791\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 30345.e have rank \(0\).
Complex multiplication
The elliptic curves in class 30345.e do not have complex multiplication.Modular form 30345.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.