Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 349830b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 349830.b4 | 349830b1 | \([1, -1, 0, -4416255, -5436237299]\) | \(-2707064176380409/2063100000000\) | \(-7259520253319100000000\) | \([2]\) | \(39911424\) | \(2.8937\) | \(\Gamma_0(N)\)-optimal |
| 349830.b3 | 349830b2 | \([1, -1, 0, -80466255, -277740867299]\) | \(16374854154935580409/4256381610000\) | \(14977116234622635210000\) | \([2, 2]\) | \(79822848\) | \(3.2403\) | |
| 349830.b2 | 349830b3 | \([1, -1, 0, -90352755, -205179889199]\) | \(23182500134142276409/8246146750089300\) | \(29016077429167150567857300\) | \([2]\) | \(159645696\) | \(3.5868\) | |
| 349830.b1 | 349830b4 | \([1, -1, 0, -1287379755, -17778710765399]\) | \(67058849150792292084409/4532630700\) | \(15949165996542062700\) | \([2]\) | \(159645696\) | \(3.5868\) |
Rank
sage: E.rank()
The elliptic curves in class 349830b have rank \(1\).
Complex multiplication
The elliptic curves in class 349830b do not have complex multiplication.Modular form 349830.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
