Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 3192.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3192.k1 | 3192o1 | \([0, 1, 0, -35, -66]\) | \(304900096/96957\) | \(1551312\) | \([2]\) | \(768\) | \(-0.10828\) | \(\Gamma_0(N)\)-optimal |
3192.k2 | 3192o2 | \([0, 1, 0, 100, -336]\) | \(427694384/477603\) | \(-122266368\) | \([2]\) | \(1536\) | \(0.23829\) |
Rank
sage: E.rank()
The elliptic curves in class 3192.k have rank \(1\).
Complex multiplication
The elliptic curves in class 3192.k do not have complex multiplication.Modular form 3192.2.a.k
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.