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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 327990bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
327990.bc1 | 327990bc1 | \([1, 1, 1, -700150, 191543027]\) | \(63812982460681/10201800960\) | \(6068269127208188160\) | \([2]\) | \(6451200\) | \(2.3265\) | \(\Gamma_0(N)\)-optimal |
327990.bc2 | 327990bc2 | \([1, 1, 1, 1250970, 1070327475]\) | \(363979050334199/1041836936400\) | \(-619708906449913784400\) | \([2]\) | \(12902400\) | \(2.6731\) |
Rank
sage: E.rank()
The elliptic curves in class 327990bc have rank \(1\).
Complex multiplication
The elliptic curves in class 327990bc do not have complex multiplication.Modular form 327990.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.