Show commands:
SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 8190m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8190.h7 | 8190m1 | \([1, -1, 0, -231525, -42591339]\) | \(1882742462388824401/11650189824000\) | \(8492988381696000\) | \([2]\) | \(73728\) | \(1.8946\) | \(\Gamma_0(N)\)-optimal |
| 8190.h6 | 8190m2 | \([1, -1, 0, -372645, 15578325]\) | \(7850236389974007121/4400862921000000\) | \(3208229069409000000\) | \([2, 2]\) | \(147456\) | \(2.2412\) | |
| 8190.h5 | 8190m3 | \([1, -1, 0, -1430325, 630090981]\) | \(443915739051786565201/21894701746029840\) | \(15961237572855753360\) | \([6]\) | \(221184\) | \(2.4439\) | |
| 8190.h4 | 8190m4 | \([1, -1, 0, -4467645, 3629825325]\) | \(13527956825588849127121/25701087819771000\) | \(18736093020613059000\) | \([2]\) | \(294912\) | \(2.5878\) | |
| 8190.h8 | 8190m5 | \([1, -1, 0, 1464435, 122496381]\) | \(476437916651992691759/284661685546875000\) | \(-207518368763671875000\) | \([2]\) | \(294912\) | \(2.5878\) | |
| 8190.h2 | 8190m6 | \([1, -1, 0, -22607145, 41378528025]\) | \(1752803993935029634719121/4599740941532100\) | \(3353211146376900900\) | \([2, 6]\) | \(442368\) | \(2.7905\) | |
| 8190.h1 | 8190m7 | \([1, -1, 0, -361714095, 2647958009895]\) | \(7179471593960193209684686321/49441793310\) | \(36043067322990\) | \([6]\) | \(884736\) | \(3.1371\) | |
| 8190.h3 | 8190m8 | \([1, -1, 0, -22329315, 42444895131]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-65538183269486324328750\) | \([6]\) | \(884736\) | \(3.1371\) |
Rank
sage: E.rank()
The elliptic curves in class 8190m have rank \(0\).
Complex multiplication
The elliptic curves in class 8190m do not have complex multiplication.Modular form 8190.2.a.m
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
