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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 25410x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25410.x3 | 25410x1 | \([1, 0, 1, -38844, 2249146]\) | \(3658671062929/880165440\) | \(1559266767051840\) | \([2]\) | \(207360\) | \(1.6264\) | \(\Gamma_0(N)\)-optimal |
| 25410.x4 | 25410x2 | \([1, 0, 1, 91836, 14167162]\) | \(48351870250991/76871856600\) | \(-136183183150152600\) | \([2]\) | \(414720\) | \(1.9729\) | |
| 25410.x1 | 25410x3 | \([1, 0, 1, -2935584, 1935685882]\) | \(1579250141304807889/41926500\) | \(74275352266500\) | \([2]\) | \(622080\) | \(2.1757\) | |
| 25410.x2 | 25410x4 | \([1, 0, 1, -2931954, 1940712706]\) | \(-1573398910560073969/8138108343750\) | \(-14417155355562093750\) | \([2]\) | \(1244160\) | \(2.5222\) |
Rank
sage: E.rank()
The elliptic curves in class 25410x have rank \(1\).
Complex multiplication
The elliptic curves in class 25410x do not have complex multiplication.Modular form 25410.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
