Show commands:
SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 3312.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3312.j1 | 3312d2 | \([0, 0, 0, -419835, 104704666]\) | \(10963069081334500/1156923\) | \(863638391808\) | \([2]\) | \(14336\) | \(1.7186\) | |
3312.j2 | 3312d1 | \([0, 0, 0, -26175, 1644478]\) | \(-10627137250000/110008287\) | \(-20530186553088\) | \([2]\) | \(7168\) | \(1.3721\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3312.j have rank \(1\).
Complex multiplication
The elliptic curves in class 3312.j do not have complex multiplication.Modular form 3312.2.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 LMFDB 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 LMFDB labels.