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SageMath
E = EllipticCurve("bv1")
E.isogeny_class()
Elliptic curves in class 333270.bv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.bv1 | 333270bv6 | \([1, -1, 0, -79984899, -275314176257]\) | \(524388516989299201/3150\) | \(339942213705150\) | \([2]\) | \(23068672\) | \(2.8538\) | |
333270.bv2 | 333270bv4 | \([1, -1, 0, -4999149, -4300678607]\) | \(128031684631201/9922500\) | \(1070817973171222500\) | \([2, 2]\) | \(11534336\) | \(2.5072\) | |
333270.bv3 | 333270bv5 | \([1, -1, 0, -4665879, -4899031565]\) | \(-104094944089921/35880468750\) | \(-3872154277985224218750\) | \([2]\) | \(23068672\) | \(2.8538\) | |
333270.bv4 | 333270bv3 | \([1, -1, 0, -1761669, 851066185]\) | \(5602762882081/345888060\) | \(37327604066850712860\) | \([2]\) | \(11534336\) | \(2.5072\) | |
333270.bv5 | 333270bv2 | \([1, -1, 0, -333369, -57618275]\) | \(37966934881/8643600\) | \(932801434406931600\) | \([2, 2]\) | \(5767168\) | \(2.1607\) | |
333270.bv6 | 333270bv1 | \([1, -1, 0, 47511, -5590067]\) | \(109902239/188160\) | \(-20305881565320960\) | \([2]\) | \(2883584\) | \(1.8141\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 333270.bv have rank \(0\).
Complex multiplication
The elliptic curves in class 333270.bv do not have complex multiplication.Modular form 333270.2.a.bv
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.