Show commands:
SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 5040o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5040.x4 | 5040o1 | \([0, 0, 0, 78, -2189]\) | \(4499456/180075\) | \(-2100394800\) | \([2]\) | \(2048\) | \(0.46863\) | \(\Gamma_0(N)\)-optimal |
5040.x3 | 5040o2 | \([0, 0, 0, -2127, -36146]\) | \(5702413264/275625\) | \(51438240000\) | \([2, 2]\) | \(4096\) | \(0.81521\) | |
5040.x1 | 5040o3 | \([0, 0, 0, -33627, -2373446]\) | \(5633270409316/14175\) | \(10581580800\) | \([2]\) | \(8192\) | \(1.1618\) | |
5040.x2 | 5040o4 | \([0, 0, 0, -5907, 127906]\) | \(30534944836/8203125\) | \(6123600000000\) | \([4]\) | \(8192\) | \(1.1618\) |
Rank
sage: E.rank()
The elliptic curves in class 5040o have rank \(1\).
Complex multiplication
The elliptic curves in class 5040o do not have complex multiplication.Modular form 5040.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.