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SageMath
E = EllipticCurve("pk1")
E.isogeny_class()
Elliptic curves in class 381150.pk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
381150.pk1 | 381150pk3 | \([1, -1, 1, -2863730, 1863684397]\) | \(4767078987/6860\) | \(3737589019034062500\) | \([2]\) | \(9953280\) | \(2.4662\) | |
381150.pk2 | 381150pk4 | \([1, -1, 1, -2046980, 2948328397]\) | \(-1740992427/5882450\) | \(-3204982583821708593750\) | \([2]\) | \(19906560\) | \(2.8128\) | |
381150.pk3 | 381150pk1 | \([1, -1, 1, -141230, -17865603]\) | \(416832723/56000\) | \(41853128625000000\) | \([2]\) | \(3317760\) | \(1.9169\) | \(\Gamma_0(N)\)-optimal |
381150.pk4 | 381150pk2 | \([1, -1, 1, 221770, -94821603]\) | \(1613964717/6125000\) | \(-4577685943359375000\) | \([2]\) | \(6635520\) | \(2.2634\) |
Rank
sage: E.rank()
The elliptic curves in class 381150.pk have rank \(0\).
Complex multiplication
The elliptic curves in class 381150.pk do not have complex multiplication.Modular form 381150.2.a.pk
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.