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SageMath
E = EllipticCurve("hc1")
E.isogeny_class()
Elliptic curves in class 162624hc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162624.bb3 | 162624hc1 | \([0, -1, 0, -3549, 82029]\) | \(2725888/21\) | \(38095647744\) | \([2]\) | \(184320\) | \(0.85933\) | \(\Gamma_0(N)\)-optimal |
| 162624.bb2 | 162624hc2 | \([0, -1, 0, -5969, -41391]\) | \(810448/441\) | \(12800137641984\) | \([2, 2]\) | \(368640\) | \(1.2059\) | |
| 162624.bb4 | 162624hc3 | \([0, -1, 0, 23071, -349215]\) | \(11696828/7203\) | \(-836275659276288\) | \([2]\) | \(737280\) | \(1.5525\) | |
| 162624.bb1 | 162624hc4 | \([0, -1, 0, -73729, -7671167]\) | \(381775972/567\) | \(65829279301632\) | \([2]\) | \(737280\) | \(1.5525\) |
Rank
sage: E.rank()
The elliptic curves in class 162624hc have rank \(1\).
Complex multiplication
The elliptic curves in class 162624hc do not have complex multiplication.Modular form 162624.2.a.hc
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.
