Show commands:
SageMath
E = EllipticCurve("gd1")
E.isogeny_class()
Elliptic curves in class 291312gd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
291312.gd2 | 291312gd1 | \([0, 0, 0, 1310037, -111967270]\) | \(3449795831/2071552\) | \(-149305856764875374592\) | \([2]\) | \(13271040\) | \(2.5603\) | \(\Gamma_0(N)\)-optimal |
291312.gd1 | 291312gd2 | \([0, 0, 0, -5348523, -904335910]\) | \(234770924809/130960928\) | \(9438929632354465087488\) | \([2]\) | \(26542080\) | \(2.9069\) |
Rank
sage: E.rank()
The elliptic curves in class 291312gd have rank \(0\).
Complex multiplication
The elliptic curves in class 291312gd do not have complex multiplication.Modular form 291312.2.a.gd
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.