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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 12705.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12705.i1 | 12705i2 | \([0, 1, 1, -101801, 12468605]\) | \(-65860951343104/3493875\) | \(-6189612688875\) | \([]\) | \(51840\) | \(1.5223\) | |
12705.i2 | 12705i1 | \([0, 1, 1, -161, 45656]\) | \(-262144/509355\) | \(-902353453155\) | \([]\) | \(17280\) | \(0.97299\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12705.i have rank \(1\).
Complex multiplication
The elliptic curves in class 12705.i do not have complex multiplication.Modular form 12705.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.