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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 4290b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4290.b4 | 4290b1 | \([1, 1, 0, -1749568, 325439488]\) | \(592265697637387401314569/296787655248366796800\) | \(296787655248366796800\) | \([2]\) | \(168960\) | \(2.6211\) | \(\Gamma_0(N)\)-optimal |
4290.b2 | 4290b2 | \([1, 1, 0, -22721088, 41643528192]\) | \(1297212465095901089487274249/1193746061037404160000\) | \(1193746061037404160000\) | \([2, 2]\) | \(337920\) | \(2.9676\) | |
4290.b1 | 4290b3 | \([1, 1, 0, -363457088, 2666878113792]\) | \(5309860874757074224246393258249/4502770931800627200\) | \(4502770931800627200\) | \([2]\) | \(675840\) | \(3.3142\) | |
4290.b3 | 4290b4 | \([1, 1, 0, -17529408, 61186050048]\) | \(-595697118196750093952139529/1272946549598037600000000\) | \(-1272946549598037600000000\) | \([2]\) | \(675840\) | \(3.3142\) |
Rank
sage: E.rank()
The elliptic curves in class 4290b have rank \(0\).
Complex multiplication
The elliptic curves in class 4290b do not have complex multiplication.Modular form 4290.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.