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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("h1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 1050.h have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(7\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 1050.h do not have complex multiplication.

Modular form 1050.2.a.h

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} + q^{9} + 4 q^{11} + q^{12} + 2 q^{13} + q^{14} + q^{16} - 2 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.

Elliptic curves in class 1050.h

Copy content 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