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SageMath
E = EllipticCurve("ic1")
E.isogeny_class()
Elliptic curves in class 206400.ic
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 206400.ic1 | 206400hc3 | \([0, 1, 0, -391233, 92901663]\) | \(1616855892553/22851963\) | \(93601640448000000\) | \([2]\) | \(1966080\) | \(2.0613\) | |
| 206400.ic2 | 206400hc2 | \([0, 1, 0, -47233, -1698337]\) | \(2845178713/1347921\) | \(5521084416000000\) | \([2, 2]\) | \(983040\) | \(1.7147\) | |
| 206400.ic3 | 206400hc1 | \([0, 1, 0, -39233, -3002337]\) | \(1630532233/1161\) | \(4755456000000\) | \([2]\) | \(491520\) | \(1.3681\) | \(\Gamma_0(N)\)-optimal |
| 206400.ic4 | 206400hc4 | \([0, 1, 0, 168767, -12714337]\) | \(129784785047/92307627\) | \(-378092040192000000\) | \([2]\) | \(1966080\) | \(2.0613\) |
Rank
sage: E.rank()
The elliptic curves in class 206400.ic have rank \(0\).
Complex multiplication
The elliptic curves in class 206400.ic do not have complex multiplication.Modular form 206400.2.a.ic
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
