Show commands:
SageMath
E = EllipticCurve("eh1")
E.isogeny_class()
Elliptic curves in class 229320.eh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
229320.eh1 | 229320em1 | \([0, 0, 0, -56742, 5194049]\) | \(397526095872/739375\) | \(37578267090000\) | \([2]\) | \(688128\) | \(1.4955\) | \(\Gamma_0(N)\)-optimal |
229320.eh2 | 229320em2 | \([0, 0, 0, -38367, 8615474]\) | \(-7680778992/34987225\) | \(-28451257579180800\) | \([2]\) | \(1376256\) | \(1.8420\) |
Rank
sage: E.rank()
The elliptic curves in class 229320.eh have rank \(1\).
Complex multiplication
The elliptic curves in class 229320.eh do not have complex multiplication.Modular form 229320.2.a.eh
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.