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Properties

Label 333270cb
Number of curves $8$
Conductor $333270$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cb1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 333270cb have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 333270cb do not have complex multiplication.

Modular form 333270.2.a.cb

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} - 4 q^{11} - 2 q^{13} - q^{14} + q^{16} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 333270cb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.cb7 333270cb1 \([1, -1, 0, 999711, 289659645]\) \(1023887723039/928972800\) \(-100253038128213196800\) \([2]\) \(11534336\) \(2.5252\) \(\Gamma_0(N)\)-optimal
333270.cb6 333270cb2 \([1, -1, 0, -5094369, 2596878333]\) \(135487869158881/51438240000\) \(5551120372919617440000\) \([2, 2]\) \(23068672\) \(2.8718\)  
333270.cb4 333270cb3 \([1, -1, 0, -71748369, 233872927533]\) \(378499465220294881/120530818800\) \(13007464559544860722800\) \([2]\) \(46137344\) \(3.2183\)  
333270.cb5 333270cb4 \([1, -1, 0, -35945649, -81090303795]\) \(47595748626367201/1215506250000\) \(131175201713474756250000\) \([2, 2]\) \(46137344\) \(3.2183\)  
333270.cb8 333270cb5 \([1, -1, 0, 6046371, -259245647847]\) \(226523624554079/269165039062500\) \(-29047796581250610351562500\) \([2]\) \(92274688\) \(3.5649\)  
333270.cb2 333270cb6 \([1, -1, 0, -571558149, -5259284831295]\) \(191342053882402567201/129708022500\) \(13997851525069017322500\) \([2, 2]\) \(92274688\) \(3.5649\)  
333270.cb3 333270cb7 \([1, -1, 0, -567987399, -5328243869445]\) \(-187778242790732059201/4984939585440150\) \(-537965523130462640865102150\) \([2]\) \(184549376\) \(3.9115\)  
333270.cb1 333270cb8 \([1, -1, 0, -9144928899, -336601202903145]\) \(783736670177727068275201/360150\) \(38866726433622150\) \([2]\) \(184549376\) \(3.9115\)