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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 345576bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 345576.bc3 | 345576bc1 | \([0, 1, 0, -14439, -672630]\) | \(11745974272/357\) | \(10119156432\) | \([2]\) | \(409600\) | \(1.0177\) | \(\Gamma_0(N)\)-optimal |
| 345576.bc2 | 345576bc2 | \([0, 1, 0, -15044, -613824]\) | \(830321872/127449\) | \(57800621539584\) | \([2, 2]\) | \(819200\) | \(1.3642\) | |
| 345576.bc1 | 345576bc3 | \([0, 1, 0, -65864, 5891136]\) | \(17418812548/1753941\) | \(3181786595226624\) | \([2]\) | \(1638400\) | \(1.7108\) | |
| 345576.bc4 | 345576bc4 | \([0, 1, 0, 26096, -3345520]\) | \(1083360092/3306177\) | \(-5997664493872128\) | \([2]\) | \(1638400\) | \(1.7108\) |
Rank
sage: E.rank()
The elliptic curves in class 345576bc have rank \(2\).
Complex multiplication
The elliptic curves in class 345576bc do not have complex multiplication.Modular form 345576.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
