Show commands:
SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 7728.t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7728.t1 | 7728p2 | \([0, 1, 0, -4004232, -3085271820]\) | \(1733490909744055732873/99355964553216\) | \(406962030809972736\) | \([2]\) | \(202752\) | \(2.4426\) | |
7728.t2 | 7728p1 | \([0, 1, 0, -235912, -54035212]\) | \(-354499561600764553/101902222098432\) | \(-417391501715177472\) | \([2]\) | \(101376\) | \(2.0961\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 7728.t have rank \(1\).
Complex multiplication
The elliptic curves in class 7728.t do not have complex multiplication.Modular form 7728.2.a.t
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.