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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 40425bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
40425.bi1 | 40425bd1 | \([0, -1, 1, -28583, -1941682]\) | \(-56197120/3267\) | \(-150140344921875\) | \([]\) | \(136080\) | \(1.4765\) | \(\Gamma_0(N)\)-optimal |
40425.bi2 | 40425bd2 | \([0, -1, 1, 155167, -3503557]\) | \(8990228480/5314683\) | \(-244244976666796875\) | \([]\) | \(408240\) | \(2.0258\) |
Rank
sage: E.rank()
The elliptic curves in class 40425bd have rank \(1\).
Complex multiplication
The elliptic curves in class 40425bd do not have complex multiplication.Modular form 40425.2.a.bd
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.