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SageMath
E = EllipticCurve("ic1")
E.isogeny_class()
Elliptic curves in class 92400.ic
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 92400.ic1 | 92400cm6 | \([0, 1, 0, -42122008, 105206239988]\) | \(258286045443018193442/8440380939375\) | \(270092190060000000000\) | \([2]\) | \(6291456\) | \(3.0144\) | |
| 92400.ic2 | 92400cm4 | \([0, 1, 0, -11900008, -15804250012]\) | \(11647843478225136004/128410942275\) | \(2054575076400000000\) | \([2]\) | \(3145728\) | \(2.6678\) | |
| 92400.ic3 | 92400cm3 | \([0, 1, 0, -2747008, 1492489988]\) | \(143279368983686884/22699269140625\) | \(363188306250000000000\) | \([2, 2]\) | \(3145728\) | \(2.6678\) | |
| 92400.ic4 | 92400cm2 | \([0, 1, 0, -762508, -234025012]\) | \(12257375872392016/1191317675625\) | \(4765270702500000000\) | \([2, 2]\) | \(1572864\) | \(2.3212\) | |
| 92400.ic5 | 92400cm1 | \([0, 1, 0, 57617, -17512012]\) | \(84611246065664/580054565475\) | \(-145013641368750000\) | \([2]\) | \(786432\) | \(1.9746\) | \(\Gamma_0(N)\)-optimal |
| 92400.ic6 | 92400cm5 | \([0, 1, 0, 4875992, 8307451988]\) | \(400647648358480318/1163177490234375\) | \(-37221679687500000000000\) | \([2]\) | \(6291456\) | \(3.0144\) |
Rank
sage: E.rank()
The elliptic curves in class 92400.ic have rank \(0\).
Complex multiplication
The elliptic curves in class 92400.ic do not have complex multiplication.Modular form 92400.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
