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SageMath
sage: E = EllipticCurve("h1")
sage: E.isogeny_class()
Elliptic curves in class 1050.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1050.h1 | 1050g5 | \([1, 0, 1, -420001, -104801602]\) | \(524388516989299201/3150\) | \(49218750\) | \([2]\) | \(6144\) | \(1.5415\) | |
| 1050.h2 | 1050g3 | \([1, 0, 1, -26251, -1639102]\) | \(128031684631201/9922500\) | \(155039062500\) | \([2, 2]\) | \(3072\) | \(1.1949\) | |
| 1050.h3 | 1050g6 | \([1, 0, 1, -24501, -1866602]\) | \(-104094944089921/35880468750\) | \(-560632324218750\) | \([2]\) | \(6144\) | \(1.5415\) | |
| 1050.h4 | 1050g4 | \([1, 0, 1, -9251, 322898]\) | \(5602762882081/345888060\) | \(5404500937500\) | \([2]\) | \(3072\) | \(1.1949\) | |
| 1050.h5 | 1050g2 | \([1, 0, 1, -1751, -22102]\) | \(37966934881/8643600\) | \(135056250000\) | \([2, 2]\) | \(1536\) | \(0.84833\) | |
| 1050.h6 | 1050g1 | \([1, 0, 1, 249, -2102]\) | \(109902239/188160\) | \(-2940000000\) | \([2]\) | \(768\) | \(0.50176\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1050.h have rank \(0\).
Complex multiplication
The elliptic curves in class 1050.h do not have complex multiplication.Modular form 1050.2.a.h
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
