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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 378.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
378.e1 | 378a3 | \([1, -1, 1, -9560, -357371]\) | \(-545407363875/14\) | \(-2480058\) | \([]\) | \(324\) | \(0.74319\) | |
378.e2 | 378a2 | \([1, -1, 1, -110, -539]\) | \(-7414875/2744\) | \(-54010152\) | \([3]\) | \(108\) | \(0.19388\) | |
378.e3 | 378a1 | \([1, -1, 1, 10, 5]\) | \(4492125/3584\) | \(-96768\) | \([3]\) | \(36\) | \(-0.35542\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 378.e have rank \(0\).
Complex multiplication
The elliptic curves in class 378.e do not have complex multiplication.Modular form 378.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.