Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 3025.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3025.a1 | 3025g3 | \([0, 1, 1, -23656508, 44278891894]\) | \(-52893159101157376/11\) | \(-304487046875\) | \([]\) | \(84000\) | \(2.5004\) | |
3025.a2 | 3025g2 | \([0, 1, 1, -31258, 3842394]\) | \(-122023936/161051\) | \(-4457994853296875\) | \([]\) | \(16800\) | \(1.6957\) | |
3025.a3 | 3025g1 | \([0, 1, 1, -1008, -29606]\) | \(-4096/11\) | \(-304487046875\) | \([]\) | \(3360\) | \(0.89094\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3025.a have rank \(0\).
Complex multiplication
The elliptic curves in class 3025.a do not have complex multiplication.Modular form 3025.2.a.a
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.
\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.