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SageMath
E = EllipticCurve("hg1")
E.isogeny_class()
Elliptic curves in class 162240.hg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162240.hg1 | 162240fa4 | \([0, 1, 0, -108385, -13770145]\) | \(890277128/15\) | \(2372473159680\) | \([2]\) | \(589824\) | \(1.5049\) | |
| 162240.hg2 | 162240fa3 | \([0, 1, 0, -27265, 1512863]\) | \(14172488/1875\) | \(296559144960000\) | \([2]\) | \(589824\) | \(1.5049\) | |
| 162240.hg3 | 162240fa2 | \([0, 1, 0, -6985, -202825]\) | \(1906624/225\) | \(4448387174400\) | \([2, 2]\) | \(294912\) | \(1.1583\) | |
| 162240.hg4 | 162240fa1 | \([0, 1, 0, 620, -15742]\) | \(85184/405\) | \(-125110889280\) | \([2]\) | \(147456\) | \(0.81170\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 162240.hg have rank \(1\).
Complex multiplication
The elliptic curves in class 162240.hg do not have complex multiplication.Modular form 162240.2.a.hg
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
