Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 33810i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33810.j2 | 33810i1 | \([1, 1, 0, -9433, 341713]\) | \(789145184521/17996580\) | \(2117279640420\) | \([2]\) | \(92160\) | \(1.1517\) | \(\Gamma_0(N)\)-optimal |
33810.j1 | 33810i2 | \([1, 1, 0, -20703, -647793]\) | \(8341959848041/3327411150\) | \(391466594386350\) | \([2]\) | \(184320\) | \(1.4983\) |
Rank
sage: E.rank()
The elliptic curves in class 33810i have rank \(1\).
Complex multiplication
The elliptic curves in class 33810i do not have complex multiplication.Modular form 33810.2.a.i
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.