Show commands:
SageMath
E = EllipticCurve("de1")
E.isogeny_class()
Elliptic curves in class 316050de
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
316050.de1 | 316050de1 | \([1, 0, 1, -234001, -40189852]\) | \(770842973809/66873600\) | \(122931440100000000\) | \([2]\) | \(4423680\) | \(2.0200\) | \(\Gamma_0(N)\)-optimal |
316050.de2 | 316050de2 | \([1, 0, 1, 255999, -186209852]\) | \(1009328859791/8734528080\) | \(-16056382720061250000\) | \([2]\) | \(8847360\) | \(2.3665\) |
Rank
sage: E.rank()
The elliptic curves in class 316050de have rank \(2\).
Complex multiplication
The elliptic curves in class 316050de do not have complex multiplication.Modular form 316050.2.a.de
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.