Show commands:
SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 333200.ex
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333200.ex1 | 333200ex1 | \([0, 1, 0, -26911208, 53727521588]\) | \(-4768951705/272\) | \(-122933228364800000000\) | \([]\) | \(11612160\) | \(2.9188\) | \(\Gamma_0(N)\)-optimal |
333200.ex2 | 333200ex2 | \([0, 1, 0, -2901208, 145109581588]\) | \(-5975305/20123648\) | \(-9095091967341363200000000\) | \([]\) | \(34836480\) | \(3.4681\) |
Rank
sage: E.rank()
The elliptic curves in class 333200.ex have rank \(1\).
Complex multiplication
The elliptic curves in class 333200.ex do not have complex multiplication.Modular form 333200.2.a.ex
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.