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SageMath
E = EllipticCurve("ct1")
E.isogeny_class()
Elliptic curves in class 25410.ct
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25410.ct1 | 25410ct8 | \([1, 0, 0, -42499135, 106635995225]\) | \(4791901410190533590281/41160000\) | \(72917450760000\) | \([2]\) | \(1658880\) | \(2.7012\) | |
25410.ct2 | 25410ct6 | \([1, 0, 0, -2656255, 1665943577]\) | \(1169975873419524361/108425318400\) | \(192082065490022400\) | \([2, 2]\) | \(829440\) | \(2.3546\) | |
25410.ct3 | 25410ct7 | \([1, 0, 0, -2462655, 1919133657]\) | \(-932348627918877961/358766164249920\) | \(-635576144704752525120\) | \([2]\) | \(1658880\) | \(2.7012\) | |
25410.ct4 | 25410ct5 | \([1, 0, 0, -527260, 144727100]\) | \(9150443179640281/184570312500\) | \(326977567382812500\) | \([2]\) | \(552960\) | \(2.1519\) | |
25410.ct5 | 25410ct3 | \([1, 0, 0, -178175, 21985305]\) | \(353108405631241/86318776320\) | \(152918977696235520\) | \([2]\) | \(414720\) | \(2.0081\) | |
25410.ct6 | 25410ct2 | \([1, 0, 0, -69880, -3738448]\) | \(21302308926361/8930250000\) | \(15820482620250000\) | \([2, 2]\) | \(276480\) | \(1.8053\) | |
25410.ct7 | 25410ct1 | \([1, 0, 0, -60200, -5688000]\) | \(13619385906841/6048000\) | \(10714400928000\) | \([2]\) | \(138240\) | \(1.4588\) | \(\Gamma_0(N)\)-optimal |
25410.ct8 | 25410ct4 | \([1, 0, 0, 232620, -27393948]\) | \(785793873833639/637994920500\) | \(-1130246919355900500\) | \([2]\) | \(552960\) | \(2.1519\) |
Rank
sage: E.rank()
The elliptic curves in class 25410.ct have rank \(1\).
Complex multiplication
The elliptic curves in class 25410.ct do not have complex multiplication.Modular form 25410.2.a.ct
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.