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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 355740.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
355740.e1 | 355740e2 | \([0, -1, 0, -12756, -515304]\) | \(5726576/405\) | \(16235336113920\) | \([2]\) | \(829440\) | \(1.2821\) | |
355740.e2 | 355740e1 | \([0, -1, 0, 719, -35594]\) | \(16384/225\) | \(-563726948400\) | \([2]\) | \(414720\) | \(0.93556\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 355740.e have rank \(1\).
Complex multiplication
The elliptic curves in class 355740.e do not have complex multiplication.Modular form 355740.2.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.