Show commands:
SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 9680t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9680.ba3 | 9680t1 | \([0, -1, 0, -161, 596]\) | \(16384/5\) | \(141724880\) | \([2]\) | \(2880\) | \(0.26900\) | \(\Gamma_0(N)\)-optimal |
9680.ba4 | 9680t2 | \([0, -1, 0, 444, 3500]\) | \(21296/25\) | \(-11337990400\) | \([2]\) | \(5760\) | \(0.61557\) | |
9680.ba1 | 9680t3 | \([0, -1, 0, -5001, -134440]\) | \(488095744/125\) | \(3543122000\) | \([2]\) | \(8640\) | \(0.81830\) | |
9680.ba2 | 9680t4 | \([0, -1, 0, -4396, -168804]\) | \(-20720464/15625\) | \(-7086244000000\) | \([2]\) | \(17280\) | \(1.1649\) |
Rank
sage: E.rank()
The elliptic curves in class 9680t have rank \(1\).
Complex multiplication
The elliptic curves in class 9680t do not have complex multiplication.Modular form 9680.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.