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SageMath
E = EllipticCurve("hg1")
E.isogeny_class()
Elliptic curves in class 47040.hg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.hg1 | 47040dn6 | \([0, 1, 0, -627265, 191007263]\) | \(1770025017602/75\) | \(1156536729600\) | \([2]\) | \(393216\) | \(1.7995\) | |
| 47040.hg2 | 47040dn4 | \([0, 1, 0, -39265, 2964863]\) | \(868327204/5625\) | \(43370127360000\) | \([2, 2]\) | \(196608\) | \(1.4530\) | |
| 47040.hg3 | 47040dn5 | \([0, 1, 0, -15745, 6506975]\) | \(-27995042/1171875\) | \(-18070886400000000\) | \([2]\) | \(393216\) | \(1.7995\) | |
| 47040.hg4 | 47040dn2 | \([0, 1, 0, -3985, -19825]\) | \(3631696/2025\) | \(3903311462400\) | \([2, 2]\) | \(98304\) | \(1.1064\) | |
| 47040.hg5 | 47040dn1 | \([0, 1, 0, -3005, -64317]\) | \(24918016/45\) | \(5421265920\) | \([2]\) | \(49152\) | \(0.75981\) | \(\Gamma_0(N)\)-optimal |
| 47040.hg6 | 47040dn3 | \([0, 1, 0, 15615, -141345]\) | \(54607676/32805\) | \(-252934582763520\) | \([2]\) | \(196608\) | \(1.4530\) |
Rank
sage: E.rank()
The elliptic curves in class 47040.hg have rank \(0\).
Complex multiplication
The elliptic curves in class 47040.hg do not have complex multiplication.Modular form 47040.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
