Show commands:
SageMath
E = EllipticCurve("hg1")
E.isogeny_class()
Elliptic curves in class 130050hg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 130050.db3 | 130050hg1 | \([1, -1, 0, -55542, 5752116]\) | \(-1860867/320\) | \(-3258571815000000\) | \([2]\) | \(884736\) | \(1.7030\) | \(\Gamma_0(N)\)-optimal |
| 130050.db2 | 130050hg2 | \([1, -1, 0, -922542, 341281116]\) | \(8527173507/200\) | \(2036607384375000\) | \([2]\) | \(1769472\) | \(2.0495\) | |
| 130050.db4 | 130050hg3 | \([1, -1, 0, 377958, -24448384]\) | \(804357/500\) | \(-3711716958023437500\) | \([2]\) | \(2654208\) | \(2.2523\) | |
| 130050.db1 | 130050hg4 | \([1, -1, 0, -1572792, -198065134]\) | \(57960603/31250\) | \(231982309876464843750\) | \([2]\) | \(5308416\) | \(2.5988\) |
Rank
sage: E.rank()
The elliptic curves in class 130050hg have rank \(1\).
Complex multiplication
The elliptic curves in class 130050hg do not have complex multiplication.Modular form 130050.2.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 Cremona 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 Cremona labels.
