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SageMath
E = EllipticCurve("eb1")
E.isogeny_class()
Elliptic curves in class 25200eb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25200.i5 | 25200eb1 | \([0, 0, 0, -14475, 1480250]\) | \(-7189057/16128\) | \(-752467968000000\) | \([2]\) | \(98304\) | \(1.5431\) | \(\Gamma_0(N)\)-optimal |
| 25200.i4 | 25200eb2 | \([0, 0, 0, -302475, 63976250]\) | \(65597103937/63504\) | \(2962842624000000\) | \([2, 2]\) | \(196608\) | \(1.8897\) | |
| 25200.i3 | 25200eb3 | \([0, 0, 0, -374475, 31216250]\) | \(124475734657/63011844\) | \(2939880593664000000\) | \([2, 2]\) | \(393216\) | \(2.2363\) | |
| 25200.i1 | 25200eb4 | \([0, 0, 0, -4838475, 4096480250]\) | \(268498407453697/252\) | \(11757312000000\) | \([2]\) | \(393216\) | \(2.2363\) | |
| 25200.i6 | 25200eb5 | \([0, 0, 0, 1389525, 241132250]\) | \(6359387729183/4218578658\) | \(-196822005867648000000\) | \([2]\) | \(786432\) | \(2.5828\) | |
| 25200.i2 | 25200eb6 | \([0, 0, 0, -3290475, -2275339750]\) | \(84448510979617/933897762\) | \(43571933983872000000\) | \([2]\) | \(786432\) | \(2.5828\) |
Rank
sage: E.rank()
The elliptic curves in class 25200eb have rank \(1\).
Complex multiplication
The elliptic curves in class 25200eb do not have complex multiplication.Modular form 25200.2.a.eb
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
