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SageMath
E = EllipticCurve("gg1")
E.isogeny_class()
Elliptic curves in class 33600gg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33600.dx4 | 33600gg1 | \([0, 1, 0, -17533, -899437]\) | \(37256083456/525\) | \(8400000000\) | \([2]\) | \(49152\) | \(1.0445\) | \(\Gamma_0(N)\)-optimal |
| 33600.dx3 | 33600gg2 | \([0, 1, 0, -18033, -845937]\) | \(2533446736/275625\) | \(70560000000000\) | \([2, 2]\) | \(98304\) | \(1.3911\) | |
| 33600.dx5 | 33600gg3 | \([0, 1, 0, 23967, -4163937]\) | \(1486779836/8203125\) | \(-8400000000000000\) | \([2]\) | \(196608\) | \(1.7376\) | |
| 33600.dx2 | 33600gg4 | \([0, 1, 0, -68033, 5904063]\) | \(34008619684/4862025\) | \(4978713600000000\) | \([2, 2]\) | \(196608\) | \(1.7376\) | |
| 33600.dx6 | 33600gg5 | \([0, 1, 0, 111967, 32004063]\) | \(75798394558/259416045\) | \(-531284060160000000\) | \([2]\) | \(393216\) | \(2.0842\) | |
| 33600.dx1 | 33600gg6 | \([0, 1, 0, -1048033, 412604063]\) | \(62161150998242/1607445\) | \(3292047360000000\) | \([4]\) | \(393216\) | \(2.0842\) |
Rank
sage: E.rank()
The elliptic curves in class 33600gg have rank \(1\).
Complex multiplication
The elliptic curves in class 33600gg do not have complex multiplication.Modular form 33600.2.a.gg
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.
