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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 13456.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13456.i1 | 13456f2 | \([0, -1, 0, -6122760, 5872930288]\) | \(-10418796526321/82044596\) | \(-199893152001324302336\) | \([]\) | \(403200\) | \(2.7240\) | |
13456.i2 | 13456f1 | \([0, -1, 0, 67000, -11109392]\) | \(13651919/29696\) | \(-72351225202343936\) | \([]\) | \(80640\) | \(1.9192\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 13456.i have rank \(0\).
Complex multiplication
The elliptic curves in class 13456.i do not have complex multiplication.Modular form 13456.2.a.i
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.