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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 3136.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3136.k1 | 3136x2 | \([0, -1, 0, -569, -5039]\) | \(406749952\) | \(50176\) | \([]\) | \(576\) | \(0.14154\) | |
3136.k2 | 3136x1 | \([0, -1, 0, -9, 1]\) | \(1792\) | \(50176\) | \([]\) | \(192\) | \(-0.40777\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3136.k have rank \(1\).
Complex multiplication
The elliptic curves in class 3136.k do not have complex multiplication.Modular form 3136.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.