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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 786.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
786.k1 | 786i1 | \([1, 1, 1, -71, -259]\) | \(39616946929/226368\) | \(226368\) | \([2]\) | \(288\) | \(-0.13103\) | \(\Gamma_0(N)\)-optimal |
786.k2 | 786i2 | \([1, 1, 1, -31, -499]\) | \(-3301293169/100082952\) | \(-100082952\) | \([2]\) | \(576\) | \(0.21554\) |
Rank
sage: E.rank()
The elliptic curves in class 786.k have rank \(0\).
Complex multiplication
The elliptic curves in class 786.k do not have complex multiplication.Modular form 786.2.a.k
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.