Properties

Label 1050.h
Number of curves $6$
Conductor $1050$
CM no
Rank $0$
Graph

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Show commands for: 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)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} + q^{9} + 4q^{11} + q^{12} + 2q^{13} + q^{14} + q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20})\)  Toggle raw display

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.