Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 720.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
720.e1 | 720d3 | \([0, 0, 0, -963, -11502]\) | \(132304644/5\) | \(3732480\) | \([2]\) | \(256\) | \(0.34709\) | |
720.e2 | 720d2 | \([0, 0, 0, -63, -162]\) | \(148176/25\) | \(4665600\) | \([2, 2]\) | \(128\) | \(0.00051877\) | |
720.e3 | 720d1 | \([0, 0, 0, -18, 27]\) | \(55296/5\) | \(58320\) | \([2]\) | \(64\) | \(-0.34606\) | \(\Gamma_0(N)\)-optimal |
720.e4 | 720d4 | \([0, 0, 0, 117, -918]\) | \(237276/625\) | \(-466560000\) | \([2]\) | \(256\) | \(0.34709\) |
Rank
sage: E.rank()
The elliptic curves in class 720.e have rank \(0\).
Complex multiplication
The elliptic curves in class 720.e do not have complex multiplication.Modular form 720.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.