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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 2475.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2475.f1 | 2475i1 | \([0, 0, 1, -210, -1229]\) | \(-56197120/3267\) | \(-59541075\) | \([]\) | \(576\) | \(0.24810\) | \(\Gamma_0(N)\)-optimal |
2475.f2 | 2475i2 | \([0, 0, 1, 1140, -2174]\) | \(8990228480/5314683\) | \(-96860097675\) | \([]\) | \(1728\) | \(0.79740\) |
Rank
sage: E.rank()
The elliptic curves in class 2475.f have rank \(1\).
Complex multiplication
The elliptic curves in class 2475.f do not have complex multiplication.Modular form 2475.2.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.