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SageMath
E = EllipticCurve("ie1")
E.isogeny_class()
Elliptic curves in class 129360.ie
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.ie1 | 129360ic3 | \([0, 1, 0, -19020640, -31935376012]\) | \(1579250141304807889/41926500\) | \(20203973830656000\) | \([2]\) | \(5971968\) | \(2.6428\) | |
| 129360.ie2 | 129360ic4 | \([0, 1, 0, -18997120, -32018269900]\) | \(-1573398910560073969/8138108343750\) | \(-3921675503754624000000\) | \([2]\) | \(11943936\) | \(2.9894\) | |
| 129360.ie3 | 129360ic1 | \([0, 1, 0, -251680, -37232140]\) | \(3658671062929/880165440\) | \(424143191451893760\) | \([2]\) | \(1990656\) | \(2.0935\) | \(\Gamma_0(N)\)-optimal |
| 129360.ie4 | 129360ic2 | \([0, 1, 0, 595040, -233332492]\) | \(48351870250991/76871856600\) | \(-37043802346018406400\) | \([2]\) | \(3981312\) | \(2.4401\) |
Rank
sage: E.rank()
The elliptic curves in class 129360.ie have rank \(0\).
Complex multiplication
The elliptic curves in class 129360.ie do not have complex multiplication.Modular form 129360.2.a.ie
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.
