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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 990b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 990.e2 | 990b1 | \([1, -1, 0, -10734, 430740]\) | \(5066026756449723/11000000\) | \(297000000\) | \([6]\) | \(1728\) | \(0.87260\) | \(\Gamma_0(N)\)-optimal |
| 990.e3 | 990b2 | \([1, -1, 0, -10614, 440748]\) | \(-4898016158612283/236328125000\) | \(-6380859375000\) | \([6]\) | \(3456\) | \(1.2192\) | |
| 990.e1 | 990b3 | \([1, -1, 0, -14109, 140165]\) | \(15781142246787/8722841600\) | \(171691691212800\) | \([2]\) | \(5184\) | \(1.4219\) | |
| 990.e4 | 990b4 | \([1, -1, 0, 55011, 1066373]\) | \(935355271080573/566899520000\) | \(-11158283252160000\) | \([2]\) | \(10368\) | \(1.7685\) |
Rank
sage: E.rank()
The elliptic curves in class 990b have rank \(0\).
Complex multiplication
The elliptic curves in class 990b do not have complex multiplication.Modular form 990.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
