Show commands:
SageMath
E = EllipticCurve("hs1")
E.isogeny_class()
Elliptic curves in class 52800.hs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
52800.hs1 | 52800hf4 | \([0, 1, 0, -234433, -43724737]\) | \(347873904937/395307\) | \(1619177472000000\) | \([2]\) | \(393216\) | \(1.8318\) | |
52800.hs2 | 52800hf2 | \([0, 1, 0, -18433, -308737]\) | \(169112377/88209\) | \(361304064000000\) | \([2, 2]\) | \(196608\) | \(1.4852\) | |
52800.hs3 | 52800hf1 | \([0, 1, 0, -10433, 403263]\) | \(30664297/297\) | \(1216512000000\) | \([2]\) | \(98304\) | \(1.1387\) | \(\Gamma_0(N)\)-optimal |
52800.hs4 | 52800hf3 | \([0, 1, 0, 69567, -2332737]\) | \(9090072503/5845851\) | \(-23944605696000000\) | \([2]\) | \(393216\) | \(1.8318\) |
Rank
sage: E.rank()
The elliptic curves in class 52800.hs have rank \(0\).
Complex multiplication
The elliptic curves in class 52800.hs do not have complex multiplication.Modular form 52800.2.a.hs
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.