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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 6171.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6171.e1 | 6171g2 | \([0, 1, 1, -7179, 231875]\) | \(-23100424192/14739\) | \(-26111037579\) | \([]\) | \(8100\) | \(0.93972\) | |
6171.e2 | 6171g1 | \([0, 1, 1, 81, 1370]\) | \(32768/459\) | \(-813146499\) | \([]\) | \(2700\) | \(0.39042\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6171.e have rank \(0\).
Complex multiplication
The elliptic curves in class 6171.e do not have complex multiplication.Modular form 6171.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.