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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 53312bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53312.g2 | 53312bd1 | \([0, 1, 0, -188225, -3750881]\) | \(23912763841/13647872\) | \(420913739970117632\) | \([2]\) | \(1032192\) | \(2.0715\) | \(\Gamma_0(N)\)-optimal |
53312.g1 | 53312bd2 | \([0, 1, 0, -2195265, -1250122721]\) | \(37936442980801/88817792\) | \(2739227698399281152\) | \([2]\) | \(2064384\) | \(2.4181\) |
Rank
sage: E.rank()
The elliptic curves in class 53312bd have rank \(1\).
Complex multiplication
The elliptic curves in class 53312bd do not have complex multiplication.Modular form 53312.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.