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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 784.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 784.b1 | 784j6 | \([0, 1, 0, -2140728, -1206278060]\) | \(2251439055699625/25088\) | \(12089663946752\) | \([2]\) | \(6912\) | \(2.0792\) | |
| 784.b2 | 784j5 | \([0, 1, 0, -133688, -18913196]\) | \(-548347731625/1835008\) | \(-884272562962432\) | \([2]\) | \(3456\) | \(1.7326\) | |
| 784.b3 | 784j4 | \([0, 1, 0, -27848, -1475468]\) | \(4956477625/941192\) | \(453551299002368\) | \([2]\) | \(2304\) | \(1.5299\) | |
| 784.b4 | 784j2 | \([0, 1, 0, -8248, 285396]\) | \(128787625/98\) | \(47225249792\) | \([2]\) | \(768\) | \(0.98059\) | |
| 784.b5 | 784j1 | \([0, 1, 0, -408, 6292]\) | \(-15625/28\) | \(-13492928512\) | \([2]\) | \(384\) | \(0.63402\) | \(\Gamma_0(N)\)-optimal |
| 784.b6 | 784j3 | \([0, 1, 0, 3512, -133260]\) | \(9938375/21952\) | \(-10578455953408\) | \([2]\) | \(1152\) | \(1.1833\) |
Rank
sage: E.rank()
The elliptic curves in class 784.b have rank \(1\).
Complex multiplication
The elliptic curves in class 784.b do not have complex multiplication.Modular form 784.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
