Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 8050b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8050.k2 | 8050b1 | \([1, 1, 0, -228000, 40064000]\) | \(83890194895342081/3958384640000\) | \(61849760000000000\) | \([2]\) | \(107520\) | \(1.9827\) | \(\Gamma_0(N)\)-optimal |
8050.k1 | 8050b2 | \([1, 1, 0, -628000, -139536000]\) | \(1753007192038126081/478174101507200\) | \(7471470336050000000\) | \([2]\) | \(215040\) | \(2.3293\) |
Rank
sage: E.rank()
The elliptic curves in class 8050b have rank \(1\).
Complex multiplication
The elliptic curves in class 8050b do not have complex multiplication.Modular form 8050.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.