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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 25350x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25350.bk3 | 25350x1 | \([1, 0, 1, -57126, 5092648]\) | \(273359449/9360\) | \(705920816250000\) | \([2]\) | \(129024\) | \(1.6211\) | \(\Gamma_0(N)\)-optimal |
| 25350.bk2 | 25350x2 | \([1, 0, 1, -141626, -13497352]\) | \(4165509529/1368900\) | \(103240919376562500\) | \([2, 2]\) | \(258048\) | \(1.9677\) | |
| 25350.bk4 | 25350x3 | \([1, 0, 1, 407624, -92589352]\) | \(99317171591/106616250\) | \(-8040879297597656250\) | \([2]\) | \(516096\) | \(2.3142\) | |
| 25350.bk1 | 25350x4 | \([1, 0, 1, -2042876, -1123827352]\) | \(12501706118329/2570490\) | \(193863504162656250\) | \([2]\) | \(516096\) | \(2.3142\) |
Rank
sage: E.rank()
The elliptic curves in class 25350x have rank \(0\).
Complex multiplication
The elliptic curves in class 25350x do not have complex multiplication.Modular form 25350.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
