Show commands:
SageMath
E = EllipticCurve("hh1")
E.isogeny_class()
Elliptic curves in class 203280.hh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 203280.hh1 | 203280ev6 | \([0, 1, 0, -2952440, -1953137100]\) | \(784478485879202/221484375\) | \(803580069600000000\) | \([2]\) | \(5242880\) | \(2.4167\) | |
| 203280.hh2 | 203280ev4 | \([0, 1, 0, -208160, -22261692]\) | \(549871953124/200930625\) | \(364503919570560000\) | \([2, 2]\) | \(2621440\) | \(2.0701\) | |
| 203280.hh3 | 203280ev2 | \([0, 1, 0, -89580, 10039500]\) | \(175293437776/4862025\) | \(2205023710982400\) | \([2, 2]\) | \(1310720\) | \(1.7235\) | |
| 203280.hh4 | 203280ev1 | \([0, 1, 0, -88975, 10185668]\) | \(2748251600896/2205\) | \(62500672080\) | \([2]\) | \(655360\) | \(1.3770\) | \(\Gamma_0(N)\)-optimal |
| 203280.hh5 | 203280ev3 | \([0, 1, 0, 19320, 32995620]\) | \(439608956/259416045\) | \(-470601060450554880\) | \([2]\) | \(2621440\) | \(2.0701\) | |
| 203280.hh6 | 203280ev5 | \([0, 1, 0, 638840, -156765292]\) | \(7947184069438/7533176175\) | \(-27331545329170790400\) | \([2]\) | \(5242880\) | \(2.4167\) |
Rank
sage: E.rank()
The elliptic curves in class 203280.hh have rank \(1\).
Complex multiplication
The elliptic curves in class 203280.hh do not have complex multiplication.Modular form 203280.2.a.hh
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
