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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 333270b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.b4 | 333270b1 | \([1, -1, 0, -8765100, 9947610000]\) | \(690080604747409/3406760000\) | \(367651281257827560000\) | \([2]\) | \(31629312\) | \(2.7938\) | \(\Gamma_0(N)\)-optimal |
333270.b3 | 333270b2 | \([1, -1, 0, -13526100, -2047253400]\) | \(2535986675931409/1450751712200\) | \(156562459867239577288200\) | \([2]\) | \(63258624\) | \(3.1404\) | |
333270.b2 | 333270b3 | \([1, -1, 0, -50376240, -130429541844]\) | \(131010595463836369/7704101562500\) | \(831412488814461914062500\) | \([2]\) | \(94887936\) | \(3.3431\) | |
333270.b1 | 333270b4 | \([1, -1, 0, -794282490, -8615870573094]\) | \(513516182162686336369/1944885031250\) | \(209888419976233281281250\) | \([2]\) | \(189775872\) | \(3.6897\) |
Rank
sage: E.rank()
The elliptic curves in class 333270b have rank \(1\).
Complex multiplication
The elliptic curves in class 333270b do not have complex multiplication.Modular form 333270.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.