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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 29040.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
29040.k1 | 29040cd2 | \([0, -1, 0, -229896, -30904080]\) | \(2711280982499089/732421875000\) | \(363000000000000000\) | \([]\) | \(311040\) | \(2.0775\) | |
29040.k2 | 29040cd1 | \([0, -1, 0, -82056, 9071856]\) | \(123286270205329/43200000\) | \(21410611200000\) | \([]\) | \(103680\) | \(1.5282\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 29040.k have rank \(1\).
Complex multiplication
The elliptic curves in class 29040.k do not have complex multiplication.Modular form 29040.2.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.