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SageMath
E = EllipticCurve("hs1")
E.isogeny_class()
Elliptic curves in class 235200hs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 235200.hs4 | 235200hs1 | \([0, -1, 0, 4492, 1230762]\) | \(85184/5625\) | \(-661775625000000\) | \([2]\) | \(884736\) | \(1.5234\) | \(\Gamma_0(N)\)-optimal |
| 235200.hs3 | 235200hs2 | \([0, -1, 0, -148633, 21290137]\) | \(48228544/2025\) | \(15247310400000000\) | \([2, 2]\) | \(1769472\) | \(1.8700\) | |
| 235200.hs1 | 235200hs3 | \([0, -1, 0, -2353633, 1390595137]\) | \(23937672968/45\) | \(2710632960000000\) | \([2]\) | \(3538944\) | \(2.2166\) | |
| 235200.hs2 | 235200hs4 | \([0, -1, 0, -393633, -66664863]\) | \(111980168/32805\) | \(1976051427840000000\) | \([2]\) | \(3538944\) | \(2.2166\) |
Rank
sage: E.rank()
The elliptic curves in class 235200hs have rank \(1\).
Complex multiplication
The elliptic curves in class 235200hs do not have complex multiplication.Modular form 235200.2.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 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.
