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SageMath
E = EllipticCurve("bq1")
E.isogeny_class()
Elliptic curves in class 13650bq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13650.bs7 | 13650bq1 | \([1, 1, 1, -85588, 14197781]\) | \(-4437543642183289/3033210136320\) | \(-47393908380000000\) | \([2]\) | \(165888\) | \(1.9002\) | \(\Gamma_0(N)\)-optimal |
| 13650.bs6 | 13650bq2 | \([1, 1, 1, -1543588, 737365781]\) | \(26031421522845051769/5797789779600\) | \(90590465306250000\) | \([2, 2]\) | \(331776\) | \(2.2468\) | |
| 13650.bs8 | 13650bq3 | \([1, 1, 1, 694037, -211751719]\) | \(2366200373628880151/2612420149248000\) | \(-40819064832000000000\) | \([2]\) | \(497664\) | \(2.4495\) | |
| 13650.bs5 | 13650bq4 | \([1, 1, 1, -1719088, 559057781]\) | \(35958207000163259449/12145729518877500\) | \(189777023732460937500\) | \([2]\) | \(663552\) | \(2.5933\) | |
| 13650.bs3 | 13650bq5 | \([1, 1, 1, -24696088, 47227585781]\) | \(106607603143751752938169/5290068420\) | \(82657319062500\) | \([2]\) | \(663552\) | \(2.5933\) | |
| 13650.bs4 | 13650bq6 | \([1, 1, 1, -3913963, -1990439719]\) | \(424378956393532177129/136231857216000000\) | \(2128622769000000000000\) | \([2, 2]\) | \(995328\) | \(2.7961\) | |
| 13650.bs1 | 13650bq7 | \([1, 1, 1, -56641963, -164076311719]\) | \(1286229821345376481036009/247265484375000000\) | \(3863523193359375000000\) | \([2]\) | \(1990656\) | \(3.1426\) | |
| 13650.bs2 | 13650bq8 | \([1, 1, 1, -24913963, 46351560281]\) | \(109454124781830273937129/3914078300576808000\) | \(61157473446512625000000\) | \([2]\) | \(1990656\) | \(3.1426\) |
Rank
sage: E.rank()
The elliptic curves in class 13650bq have rank \(0\).
Complex multiplication
The elliptic curves in class 13650bq do not have complex multiplication.Modular form 13650.2.a.bq
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.
