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Properties

Label 130050em
Number of curves $8$
Conductor $130050$
CM no
Rank $2$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("em1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 130050em have rank \(2\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 3 T + 7 T^{2}\) 1.7.d
\(11\) \( 1 - 3 T + 11 T^{2}\) 1.11.ad
\(13\) \( 1 + 3 T + 13 T^{2}\) 1.13.d
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 5 T + 23 T^{2}\) 1.23.f
\(29\) \( 1 + 8 T + 29 T^{2}\) 1.29.i
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 130050em do not have complex multiplication.

Modular form 130050.2.a.em

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 4 q^{7} - q^{8} - 2 q^{13} + 4 q^{14} + q^{16} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 130050em

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
130050.e8 130050em1 \([1, -1, 0, 96183, -35278659]\) \(357911/2160\) \(-593874713283750000\) \([2]\) \(1769472\) \(2.0923\) \(\Gamma_0(N)\)-optimal
130050.e6 130050em2 \([1, -1, 0, -1204317, -460542159]\) \(702595369/72900\) \(20043271573326562500\) \([2, 2]\) \(3538944\) \(2.4389\)  
130050.e7 130050em3 \([1, -1, 0, -879192, 1038609216]\) \(-273359449/1536000\) \(-422310907224000000000\) \([2]\) \(5308416\) \(2.6417\)  
130050.e5 130050em4 \([1, -1, 0, -4455567, 3119084091]\) \(35578826569/5314410\) \(1461154497695506406250\) \([2]\) \(7077888\) \(2.7855\)  
130050.e4 130050em5 \([1, -1, 0, -18761067, -31272638409]\) \(2656166199049/33750\) \(9279292395058593750\) \([2]\) \(7077888\) \(2.7855\)  
130050.e3 130050em6 \([1, -1, 0, -21687192, 38805129216]\) \(4102915888729/9000000\) \(2474477972015625000000\) \([2, 2]\) \(10616832\) \(2.9882\)  
130050.e1 130050em7 \([1, -1, 0, -346812192, 2486021004216]\) \(16778985534208729/81000\) \(22270301748140625000\) \([2]\) \(21233664\) \(3.3348\)  
130050.e2 130050em8 \([1, -1, 0, -29490192, 8396838216]\) \(10316097499609/5859375000\) \(1610988263031005859375000\) \([2]\) \(21233664\) \(3.3348\)