Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 179520.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 179520.b1 | 179520hr8 | \([0, -1, 0, -12879134881, -562567756120799]\) | \(901247067798311192691198986281/552431869440\) | \(144816699982479360\) | \([2]\) | \(127401984\) | \(4.0005\) | |
| 179520.b2 | 179520hr7 | \([0, -1, 0, -810352801, -8665841992415]\) | \(224494757451893010998773801/6152490825146276160000\) | \(1612838554867145417687040000\) | \([2]\) | \(127401984\) | \(4.0005\) | |
| 179520.b3 | 179520hr6 | \([0, -1, 0, -804946081, -8789916484319]\) | \(220031146443748723000125481/172266701724057600\) | \(45158682256751355494400\) | \([2, 2]\) | \(63700992\) | \(3.6539\) | |
| 179520.b4 | 179520hr5 | \([0, -1, 0, -159034081, -771320361119]\) | \(1696892787277117093383481/1440538624914939000\) | \(377628557289701769216000\) | \([2]\) | \(42467328\) | \(3.4512\) | |
| 179520.b5 | 179520hr4 | \([0, -1, 0, -104152801, 404787487585]\) | \(476646772170172569823801/5862293314453125000\) | \(1536765018624000000000000\) | \([2]\) | \(42467328\) | \(3.4512\) | |
| 179520.b6 | 179520hr3 | \([0, -1, 0, -49971361, -139265147615]\) | \(-52643812360427830814761/1504091705903677440\) | \(-394288616152413618831360\) | \([2]\) | \(31850496\) | \(3.3074\) | |
| 179520.b7 | 179520hr2 | \([0, -1, 0, -12154081, -6281193119]\) | \(757443433548897303481/373234243041000000\) | \(97841117407739904000000\) | \([2, 2]\) | \(21233664\) | \(3.1046\) | |
| 179520.b8 | 179520hr1 | \([0, -1, 0, 2775839, -754136735]\) | \(9023321954633914439/6156756739584000\) | \(-1613956838741508096000\) | \([2]\) | \(10616832\) | \(2.7581\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 179520.b have rank \(1\).
Complex multiplication
The elliptic curves in class 179520.b do not have complex multiplication.Modular form 179520.2.a.b
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 3 & 12 & 4 & 6 & 12 \\ 4 & 1 & 2 & 12 & 3 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 3 & 12 & 6 & 1 & 4 & 12 & 2 & 4 \\ 12 & 3 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
