Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 2070.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2070.e1 | 2070h2 | \([1, -1, 0, -1044, -12200]\) | \(172715635009/7935000\) | \(5784615000\) | \([2]\) | \(1536\) | \(0.63484\) | |
2070.e2 | 2070h1 | \([1, -1, 0, 36, -752]\) | \(6967871/331200\) | \(-241444800\) | \([2]\) | \(768\) | \(0.28826\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2070.e have rank \(1\).
Complex multiplication
The elliptic curves in class 2070.e do not have complex multiplication.Modular form 2070.2.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.