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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 20400ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20400.g3 | 20400ca1 | \([0, -1, 0, -40408, -2716688]\) | \(114013572049/15667200\) | \(1002700800000000\) | \([2]\) | \(110592\) | \(1.6053\) | \(\Gamma_0(N)\)-optimal |
| 20400.g2 | 20400ca2 | \([0, -1, 0, -168408, 23907312]\) | \(8253429989329/936360000\) | \(59927040000000000\) | \([2, 2]\) | \(221184\) | \(1.9518\) | |
| 20400.g1 | 20400ca3 | \([0, -1, 0, -2616408, 1629795312]\) | \(30949975477232209/478125000\) | \(30600000000000000\) | \([2]\) | \(442368\) | \(2.2984\) | |
| 20400.g4 | 20400ca4 | \([0, -1, 0, 231592, 119907312]\) | \(21464092074671/109596256200\) | \(-7014160396800000000\) | \([2]\) | \(442368\) | \(2.2984\) |
Rank
sage: E.rank()
The elliptic curves in class 20400ca have rank \(0\).
Complex multiplication
The elliptic curves in class 20400ca do not have complex multiplication.Modular form 20400.2.a.ca
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.
