Show commands:
SageMath
E = EllipticCurve("eq1")
E.isogeny_class()
Elliptic curves in class 414050.eq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414050.eq1 | 414050eq2 | \([1, 0, 0, -93828043, 349814173617]\) | \(-6434774386429585/140608\) | \(-1996173994774523200\) | \([]\) | \(42456960\) | \(3.0378\) | \(\Gamma_0(N)\)-optimal* |
414050.eq2 | 414050eq1 | \([1, 0, 0, -1080843, 546767857]\) | \(-9836106385/3407872\) | \(-48380643092286668800\) | \([]\) | \(14152320\) | \(2.4885\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 414050.eq have rank \(1\).
Complex multiplication
The elliptic curves in class 414050.eq do not have complex multiplication.Modular form 414050.2.a.eq
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.