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SageMath
E = EllipticCurve("kc1")
E.isogeny_class()
Elliptic curves in class 58800.kc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
58800.kc1 | 58800jd2 | \([0, 1, 0, -37508, 1756488]\) | \(4253563312/1476225\) | \(2025380700000000\) | \([2]\) | \(368640\) | \(1.6387\) | |
58800.kc2 | 58800jd1 | \([0, 1, 0, -15633, -737262]\) | \(4927700992/151875\) | \(13023281250000\) | \([2]\) | \(184320\) | \(1.2921\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 58800.kc have rank \(0\).
Complex multiplication
The elliptic curves in class 58800.kc do not have complex multiplication.Modular form 58800.2.a.kc
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.