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SageMath
E = EllipticCurve("et1")
E.isogeny_class()
Elliptic curves in class 25200.et
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25200.et1 | 25200cy4 | \([0, 0, 0, -451575, 116795250]\) | \(129348709488/6125\) | \(482233500000000\) | \([2]\) | \(165888\) | \(1.8913\) | |
| 25200.et2 | 25200cy3 | \([0, 0, 0, -29700, 1623375]\) | \(588791808/109375\) | \(538207031250000\) | \([2]\) | \(82944\) | \(1.5448\) | |
| 25200.et3 | 25200cy2 | \([0, 0, 0, -10575, -167750]\) | \(1210991472/588245\) | \(63530460000000\) | \([2]\) | \(55296\) | \(1.3420\) | |
| 25200.et4 | 25200cy1 | \([0, 0, 0, -8700, -312125]\) | \(10788913152/8575\) | \(57881250000\) | \([2]\) | \(27648\) | \(0.99547\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 25200.et have rank \(1\).
Complex multiplication
The elliptic curves in class 25200.et do not have complex multiplication.Modular form 25200.2.a.et
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.
