Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 111090.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
111090.d1 | 111090d2 | \([1, 1, 0, -55037193073, -4950836077262123]\) | \(10236276715651320972447503/45003627643509964800\) | \(81058403705597812078481446502400\) | \([2]\) | \(593510400\) | \(4.9753\) | |
111090.d2 | 111090d1 | \([1, 1, 0, -5201161073, 10311006718677]\) | \(8639211347488146591503/4974033166663680000\) | \(8958993076341521080217763840000\) | \([2]\) | \(296755200\) | \(4.6288\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 111090.d have rank \(1\).
Complex multiplication
The elliptic curves in class 111090.d do not have complex multiplication.Modular form 111090.2.a.d
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.