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SageMath
E = EllipticCurve("ga1")
E.isogeny_class()
Elliptic curves in class 411840ga
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
411840.ga3 | 411840ga1 | \([0, 0, 0, -38028, -2064112]\) | \(31824875809/8785920\) | \(1679015458897920\) | \([2]\) | \(1769472\) | \(1.6289\) | \(\Gamma_0(N)\)-optimal* |
411840.ga2 | 411840ga2 | \([0, 0, 0, -222348, 38707472]\) | \(6361447449889/294465600\) | \(56273252489625600\) | \([2, 2]\) | \(3538944\) | \(1.9755\) | \(\Gamma_0(N)\)-optimal* |
411840.ga1 | 411840ga3 | \([0, 0, 0, -3517068, 2538741008]\) | \(25176685646263969/57915000\) | \(11067728855040000\) | \([2]\) | \(7077888\) | \(2.3220\) | \(\Gamma_0(N)\)-optimal* |
411840.ga4 | 411840ga4 | \([0, 0, 0, 123252, 148055312]\) | \(1083523132511/50179392120\) | \(-9589431168002949120\) | \([2]\) | \(7077888\) | \(2.3220\) |
Rank
sage: E.rank()
The elliptic curves in class 411840ga have rank \(0\).
Complex multiplication
The elliptic curves in class 411840ga do not have complex multiplication.Modular form 411840.2.a.ga
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.