Show commands:
SageMath
E = EllipticCurve("en1")
E.isogeny_class()
Elliptic curves in class 68400en
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68400.cb2 | 68400en1 | \([0, 0, 0, -282675, -202900750]\) | \(-53540005609/350208000\) | \(-16339304448000000000\) | \([2]\) | \(1548288\) | \(2.3701\) | \(\Gamma_0(N)\)-optimal |
68400.cb1 | 68400en2 | \([0, 0, 0, -7194675, -7412116750]\) | \(882774443450089/2166000000\) | \(101056896000000000000\) | \([2]\) | \(3096576\) | \(2.7167\) |
Rank
sage: E.rank()
The elliptic curves in class 68400en have rank \(1\).
Complex multiplication
The elliptic curves in class 68400en do not have complex multiplication.Modular form 68400.2.a.en
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.