Show commands:
SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 176890ef
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
176890.bt2 | 176890ef1 | \([1, 1, 0, -25638, -1868908]\) | \(-115501303/25600\) | \(-413100471884800\) | \([2]\) | \(1048320\) | \(1.5250\) | \(\Gamma_0(N)\)-optimal |
176890.bt1 | 176890ef2 | \([1, 1, 0, -429958, -108690252]\) | \(544737993463/20000\) | \(322734743660000\) | \([2]\) | \(2096640\) | \(1.8716\) |
Rank
sage: E.rank()
The elliptic curves in class 176890ef have rank \(1\).
Complex multiplication
The elliptic curves in class 176890ef do not have complex multiplication.Modular form 176890.2.a.ef
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.