Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 1694.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1694.i1 | 1694i2 | \([1, 1, 1, -28377, 1828091]\) | \(1426487591593/2156\) | \(3819485516\) | \([2]\) | \(3840\) | \(1.1066\) | |
1694.i2 | 1694i1 | \([1, 1, 1, -1757, 28579]\) | \(-338608873/13552\) | \(-24008194672\) | \([2]\) | \(1920\) | \(0.76006\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1694.i have rank \(0\).
Complex multiplication
The elliptic curves in class 1694.i do not have complex multiplication.Modular form 1694.2.a.i
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.