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Properties

Label 162450.i
Number of curves $4$
Conductor $162450$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("i1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 162450.i have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(11\) \( 1 + 6 T + 11 T^{2}\) 1.11.g
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 162450.i do not have complex multiplication.

Modular form 162450.2.a.i

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2 q^{7} - q^{8} - 6 q^{11} - 4 q^{13} + 2 q^{14} + q^{16} - 6 q^{17} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 162450.i

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162450.i1 162450dg4 \([1, -1, 0, -37625939667, 1793801850972741]\) \(10993009831928446009969/3767761230468750000\) \(2019075379493731498718261718750000\) \([2]\) \(1194393600\) \(5.0926\)  
162450.i2 162450dg2 \([1, -1, 0, -33707645667, 2382002636910741]\) \(7903870428425797297009/886464000000\) \(475040090845899000000000000\) \([2]\) \(398131200\) \(4.5433\)  
162450.i3 162450dg1 \([1, -1, 0, -2101373667, 37417773678741]\) \(-1914980734749238129/20440940544000\) \(-10953931860738146304000000000\) \([2]\) \(199065600\) \(4.1967\) \(\Gamma_0(N)\)-optimal
162450.i4 162450dg3 \([1, -1, 0, 6943842333, 194771782158741]\) \(69096190760262356111/70568821500000000\) \(-37816560374981985210937500000000\) \([2]\) \(597196800\) \(4.7460\)