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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 139230.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 139230.j1 | 139230dy7 | \([1, -1, 0, -20326751460, 1115454997162890]\) | \(1274090022584975661628188489514561/14072533302105480763470\) | \(10258876777234895476569630\) | \([2]\) | \(201326592\) | \(4.3687\) | |
| 139230.j2 | 139230dy5 | \([1, -1, 0, -1271435310, 17400026145600]\) | \(311802066473807207098058600161/1033693082103011001480900\) | \(753562256853095020079576100\) | \([2, 2]\) | \(100663296\) | \(4.0221\) | |
| 139230.j3 | 139230dy4 | \([1, -1, 0, -1251138330, -17033273811324]\) | \(297106512928238351998640242081/3977028808593750000\) | \(2899254001464843750000\) | \([2]\) | \(50331648\) | \(3.6755\) | |
| 139230.j4 | 139230dy8 | \([1, -1, 0, -723647160, 32492356581510]\) | \(-57487943130312093140621093761/592356094985924086700006670\) | \(-431827593244738659204304862430\) | \([2]\) | \(201326592\) | \(4.3687\) | |
| 139230.j5 | 139230dy3 | \([1, -1, 0, -114714810, 6882675300]\) | \(229010110533436633465952161/132501160769452503210000\) | \(96593346200930874840090000\) | \([2, 2]\) | \(50331648\) | \(3.6755\) | |
| 139230.j6 | 139230dy2 | \([1, -1, 0, -78264810, -265639394700]\) | \(72727020009972527154752161/265361167808100000000\) | \(193448291332104900000000\) | \([2, 2]\) | \(25165824\) | \(3.3289\) | |
| 139230.j7 | 139230dy1 | \([1, -1, 0, -2682090, -7917436044]\) | \(-2926956820564562516641/35459588343029760000\) | \(-25850039902068695040000\) | \([2]\) | \(12582912\) | \(2.9824\) | \(\Gamma_0(N)\)-optimal |
| 139230.j8 | 139230dy6 | \([1, -1, 0, 458805690, 54714285000]\) | \(14651516183052242700771495839/8480668142378708755560900\) | \(-6182407075794078682803896100\) | \([2]\) | \(100663296\) | \(4.0221\) |
Rank
sage: E.rank()
The elliptic curves in class 139230.j have rank \(1\).
Complex multiplication
The elliptic curves in class 139230.j do not have complex multiplication.Modular form 139230.2.a.j
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 16 & 4 & 4 & 8 & 16 & 8 \\ 2 & 1 & 8 & 2 & 2 & 4 & 8 & 4 \\ 16 & 8 & 1 & 16 & 4 & 2 & 4 & 8 \\ 4 & 2 & 16 & 1 & 4 & 8 & 16 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 8 & 2 & 1 & 2 & 4 \\ 16 & 8 & 4 & 16 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
