Show commands:
SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 33930l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33930.a4 | 33930l1 | \([1, -1, 0, 310635, -114410075]\) | \(4547226203385942959/10377808593750000\) | \(-7565422464843750000\) | \([2]\) | \(829440\) | \(2.3066\) | \(\Gamma_0(N)\)-optimal |
33930.a3 | 33930l2 | \([1, -1, 0, -2501865, -1254597575]\) | \(2375679751819859057041/441134740310062500\) | \(321587225686035562500\) | \([2, 2]\) | \(1658880\) | \(2.6532\) | |
33930.a2 | 33930l3 | \([1, -1, 0, -11963115, 14778436675]\) | \(259734139401368855237041/20937966860481050250\) | \(15263777841290685632250\) | \([2]\) | \(3317760\) | \(2.9997\) | |
33930.a1 | 33930l4 | \([1, -1, 0, -38040615, -90293381825]\) | \(8351005675201800382877041/395069604635949750\) | \(288005741779607367750\) | \([2]\) | \(3317760\) | \(2.9997\) |
Rank
sage: E.rank()
The elliptic curves in class 33930l have rank \(0\).
Complex multiplication
The elliptic curves in class 33930l do not have complex multiplication.Modular form 33930.2.a.l
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.