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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 54390.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54390.f1 | 54390h2 | \([1, 1, 0, -357473, 66590133]\) | \(103100142486286723561/20395591729152000\) | \(999383994728448000\) | \([]\) | \(1244160\) | \(2.1702\) | |
54390.f2 | 54390h1 | \([1, 1, 0, -108098, -13716492]\) | \(2850932179613533561/1998000000000\) | \(97902000000000\) | \([]\) | \(414720\) | \(1.6209\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 54390.f have rank \(1\).
Complex multiplication
The elliptic curves in class 54390.f do not have complex multiplication.Modular form 54390.2.a.f
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.