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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 32340.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32340.s1 | 32340s4 | \([0, -1, 0, -12249820, 16506315400]\) | \(6749703004355978704/5671875\) | \(170826348000000\) | \([2]\) | \(622080\) | \(2.4665\) | |
32340.s2 | 32340s3 | \([0, -1, 0, -765445, 258221650]\) | \(-26348629355659264/24169921875\) | \(-45497074218750000\) | \([2]\) | \(311040\) | \(2.1199\) | |
32340.s3 | 32340s2 | \([0, -1, 0, -154660, 21608392]\) | \(13584145739344/1195803675\) | \(36015387279379200\) | \([2]\) | \(207360\) | \(1.9172\) | |
32340.s4 | 32340s1 | \([0, -1, 0, 10715, 1564942]\) | \(72268906496/606436875\) | \(-1141547070510000\) | \([2]\) | \(103680\) | \(1.5706\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 32340.s have rank \(1\).
Complex multiplication
The elliptic curves in class 32340.s do not have complex multiplication.Modular form 32340.2.a.s
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.