Show commands:
SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 6720.u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6720.u1 | 6720bn4 | \([0, -1, 0, -14945, 708225]\) | \(5633270409316/14175\) | \(928972800\) | \([2]\) | \(8192\) | \(0.95905\) | |
6720.u2 | 6720bn3 | \([0, -1, 0, -2625, -37023]\) | \(30534944836/8203125\) | \(537600000000\) | \([2]\) | \(8192\) | \(0.95905\) | |
6720.u3 | 6720bn2 | \([0, -1, 0, -945, 11025]\) | \(5702413264/275625\) | \(4515840000\) | \([2, 2]\) | \(4096\) | \(0.61247\) | |
6720.u4 | 6720bn1 | \([0, -1, 0, 35, 637]\) | \(4499456/180075\) | \(-184396800\) | \([2]\) | \(2048\) | \(0.26590\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6720.u have rank \(1\).
Complex multiplication
The elliptic curves in class 6720.u do not have complex multiplication.Modular form 6720.2.a.u
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.