Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 435344.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
435344.e1 | 435344e2 | \([0, 1, 0, -451624, -110706348]\) | \(1030541881826/62236321\) | \(615225004707203072\) | \([2]\) | \(4147200\) | \(2.1658\) | \(\Gamma_0(N)\)-optimal* |
435344.e2 | 435344e1 | \([0, 1, 0, -444864, -114354044]\) | \(1969910093092/7889\) | \(38992584909824\) | \([2]\) | \(2073600\) | \(1.8193\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 435344.e have rank \(2\).
Complex multiplication
The elliptic curves in class 435344.e do not have complex multiplication.Modular form 435344.2.a.e
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.