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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 8190n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8190.i5 | 8190n1 | \([1, -1, 0, -3330, -1164524]\) | \(-5602762882081/801531494400\) | \(-584316459417600\) | \([2]\) | \(49152\) | \(1.5129\) | \(\Gamma_0(N)\)-optimal |
| 8190.i4 | 8190n2 | \([1, -1, 0, -187650, -30987500]\) | \(1002404925316922401/9348917760000\) | \(6815361047040000\) | \([2, 2]\) | \(98304\) | \(1.8594\) | |
| 8190.i2 | 8190n3 | \([1, -1, 0, -2995650, -1994902700]\) | \(4078208988807294650401/359723582400\) | \(262238491569600\) | \([2]\) | \(196608\) | \(2.2060\) | |
| 8190.i3 | 8190n4 | \([1, -1, 0, -328770, 22045396]\) | \(5391051390768345121/2833965225000000\) | \(2065960649025000000\) | \([2, 2]\) | \(196608\) | \(2.2060\) | |
| 8190.i1 | 8190n5 | \([1, -1, 0, -4161690, 3265462300]\) | \(10934663514379917006241/12996826171875000\) | \(9474686279296875000\) | \([2]\) | \(393216\) | \(2.5526\) | |
| 8190.i6 | 8190n6 | \([1, -1, 0, 1246230, 171040396]\) | \(293623352309352854879/187320324116835000\) | \(-136556516281172715000\) | \([2]\) | \(393216\) | \(2.5526\) |
Rank
sage: E.rank()
The elliptic curves in class 8190n have rank \(0\).
Complex multiplication
The elliptic curves in class 8190n do not have complex multiplication.Modular form 8190.2.a.n
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.
