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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 57330bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
57330.i1 | 57330bc1 | \([1, -1, 0, -1350540, 604501456]\) | \(-7626453723007966609/921488588800\) | \(-32916493880524800\) | \([]\) | \(912384\) | \(2.1938\) | \(\Gamma_0(N)\)-optimal |
57330.i2 | 57330bc2 | \([1, -1, 0, 181620, 1872533200]\) | \(18547687612920431/42417997492000000\) | \(-1515213288411732000000\) | \([]\) | \(2737152\) | \(2.7431\) |
Rank
sage: E.rank()
The elliptic curves in class 57330bc have rank \(1\).
Complex multiplication
The elliptic curves in class 57330bc do not have complex multiplication.Modular form 57330.2.a.bc
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.