Show commands:
SageMath
E = EllipticCurve("hl1")
E.isogeny_class()
Elliptic curves in class 705600.hl
sage: E.isogeny_class().curves
| LMFDB label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height |
|---|---|---|---|---|---|---|
| 705600.hl1 | \([0, 0, 0, -296366700, -1963777214000]\) | \(32779037733124/315\) | \(27664719989760000000\) | \([2]\) | \(75497472\) | \(3.3089\) |
| 705600.hl2 | \([0, 0, 0, -285782700, 1853236546000]\) | \(14695548366242/57421875\) | \(10086095829600000000000000\) | \([2]\) | \(150994944\) | \(3.6555\) |
| 705600.hl3 | \([0, 0, 0, -26474700, -1852886000]\) | \(23366901604/13505625\) | \(1186124869560960000000000\) | \([2, 2]\) | \(75497472\) | \(3.3089\) |
| 705600.hl4 | \([0, 0, 0, -18536700, -30636074000]\) | \(32082281296/99225\) | \(2178596699193600000000\) | \([2, 2]\) | \(37748736\) | \(2.9623\) |
| 705600.hl5 | \([0, 0, 0, -676200, -880481000]\) | \(-24918016/229635\) | \(-315118451133360000000\) | \([2]\) | \(18874368\) | \(2.6157\) |
| 705600.hl6 | \([0, 0, 0, 105825300, -14818286000]\) | \(746185003198/432360075\) | \(-75943621648423065600000000\) | \([2]\) | \(150994944\) | \(3.6555\) |
Rank
sage: E.rank()
The elliptic curves in class 705600.hl have rank \(0\).
Complex multiplication
The elliptic curves in class 705600.hl do not have complex multiplication.Modular form 705600.2.a.hl
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
