Show commands:
SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 33810j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33810.k1 | 33810j1 | \([1, 1, 0, -1469633, -114164427]\) | \(8698983351369247/4896098136000\) | \(197575220013576552000\) | \([2]\) | \(1354752\) | \(2.5845\) | \(\Gamma_0(N)\)-optimal |
33810.k2 | 33810j2 | \([1, 1, 0, 5788247, -899467043]\) | \(531474461802274913/316273825125000\) | \(-12762789643480975875000\) | \([2]\) | \(2709504\) | \(2.9311\) |
Rank
sage: E.rank()
The elliptic curves in class 33810j have rank \(1\).
Complex multiplication
The elliptic curves in class 33810j do not have complex multiplication.Modular form 33810.2.a.j
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.