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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 129360u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.b5 | 129360u1 | \([0, -1, 0, -51711, 4037286]\) | \(8124052043776/992578125\) | \(1868413181250000\) | \([2]\) | \(786432\) | \(1.6606\) | \(\Gamma_0(N)\)-optimal |
| 129360.b4 | 129360u2 | \([0, -1, 0, -204836, -31426464]\) | \(31558509702736/4035425625\) | \(121539530075040000\) | \([2, 2]\) | \(1572864\) | \(2.0072\) | |
| 129360.b6 | 129360u3 | \([0, -1, 0, 309664, -164373264]\) | \(27258770992316/112538412525\) | \(-13557792455837414400\) | \([2]\) | \(3145728\) | \(2.3537\) | |
| 129360.b2 | 129360u4 | \([0, -1, 0, -3169336, -2170609664]\) | \(29224056825643684/588305025\) | \(70874621835494400\) | \([2, 2]\) | \(3145728\) | \(2.3537\) | |
| 129360.b3 | 129360u5 | \([0, -1, 0, -3061536, -2325237984]\) | \(-13171152353214242/2080257264855\) | \(-501227902879592232960\) | \([2]\) | \(6291456\) | \(2.7003\) | |
| 129360.b1 | 129360u6 | \([0, -1, 0, -50709136, -138971138144]\) | \(59850000883110493442/24255\) | \(5844124661760\) | \([2]\) | \(6291456\) | \(2.7003\) |
Rank
sage: E.rank()
The elliptic curves in class 129360u have rank \(0\).
Complex multiplication
The elliptic curves in class 129360u do not have complex multiplication.Modular form 129360.2.a.u
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
