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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 111573k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
111573.l1 | 111573k1 | \([1, -1, 1, -5081, 138352]\) | \(13312053/253\) | \(275655489417\) | \([2]\) | \(118272\) | \(0.98863\) | \(\Gamma_0(N)\)-optimal |
111573.l2 | 111573k2 | \([1, -1, 1, 64, 401776]\) | \(27/64009\) | \(-69740838822501\) | \([2]\) | \(236544\) | \(1.3352\) |
Rank
sage: E.rank()
The elliptic curves in class 111573k have rank \(1\).
Complex multiplication
The elliptic curves in class 111573k do not have complex multiplication.Modular form 111573.2.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 Cremona 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 Cremona labels.