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Properties

Label 1650h
Number of curves $6$
Conductor $1650$
CM no
Rank $1$
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 1650h have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 1650h do not have complex multiplication.

Modular form 1650.2.a.h

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 1650h

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1650.h6 1650h1 \([1, 0, 1, 6374, 19148]\) \(1833318007919/1070530560\) \(-16727040000000\) \([2]\) \(4608\) \(1.2268\) \(\Gamma_0(N)\)-optimal
1650.h5 1650h2 \([1, 0, 1, -25626, 147148]\) \(119102750067601/68309049600\) \(1067328900000000\) \([2, 2]\) \(9216\) \(1.5733\)  
1650.h3 1650h3 \([1, 0, 1, -267626, -53092852]\) \(135670761487282321/643043610000\) \(10047556406250000\) \([2, 2]\) \(18432\) \(1.9199\)  
1650.h2 1650h4 \([1, 0, 1, -295626, 61707148]\) \(182864522286982801/463015182960\) \(7234612233750000\) \([2]\) \(18432\) \(1.9199\)  
1650.h1 1650h5 \([1, 0, 1, -4277126, -3405034852]\) \(553808571467029327441/12529687500\) \(195776367187500\) \([2]\) \(36864\) \(2.2665\)  
1650.h4 1650h6 \([1, 0, 1, -130126, -107542852]\) \(-15595206456730321/310672490129100\) \(-4854257658267187500\) \([2]\) \(36864\) \(2.2665\)