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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 12870.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12870.f1 | 12870k5 | \([1, -1, 0, -53539200, 150797618136]\) | \(23281546263261052473907201/1115400\) | \(813126600\) | \([2]\) | \(491520\) | \(2.6003\) | |
| 12870.f2 | 12870k3 | \([1, -1, 0, -3346200, 2356839936]\) | \(5683972151443376419201/1244117160000\) | \(906961409640000\) | \([2, 2]\) | \(245760\) | \(2.2537\) | |
| 12870.f3 | 12870k6 | \([1, -1, 0, -3334320, 2374396200]\) | \(-5623647484692626737921/84122230603125000\) | \(-61325106109678125000\) | \([2]\) | \(491520\) | \(2.6003\) | |
| 12870.f4 | 12870k2 | \([1, -1, 0, -209880, 36590400]\) | \(1402524686897642881/20523074457600\) | \(14961321279590400\) | \([2, 2]\) | \(122880\) | \(1.9071\) | |
| 12870.f5 | 12870k1 | \([1, -1, 0, -25560, -679104]\) | \(2533309721804161/1187575234560\) | \(865742345994240\) | \([2]\) | \(61440\) | \(1.5605\) | \(\Gamma_0(N)\)-optimal |
| 12870.f6 | 12870k4 | \([1, -1, 0, -22680, 99377280]\) | \(-1769848555063681/5850659851882560\) | \(-4265131032022386240\) | \([2]\) | \(245760\) | \(2.2537\) |
Rank
sage: E.rank()
The elliptic curves in class 12870.f have rank \(1\).
Complex multiplication
The elliptic curves in class 12870.f do not have complex multiplication.Modular form 12870.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
