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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 27225.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 27225.bc1 | 27225bz1 | \([0, 0, 1, -635250, 204433281]\) | \(-56197120/3267\) | \(-1648135099501171875\) | \([]\) | \(345600\) | \(2.2518\) | \(\Gamma_0(N)\)-optimal |
| 27225.bc2 | 27225bz2 | \([0, 0, 1, 3448500, 361657656]\) | \(8990228480/5314683\) | \(-2681149554644073046875\) | \([]\) | \(1036800\) | \(2.8011\) |
Rank
sage: E.rank()
The elliptic curves in class 27225.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 27225.bc do not have complex multiplication.Modular form 27225.2.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 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.
