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Properties

Label 303450cw
Number of curves $8$
Conductor $303450$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cw1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 303450cw have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 303450cw do not have complex multiplication.

Modular form 303450.2.a.cw

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 303450cw

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
303450.cw7 303450cw1 \([1, 0, 1, -233555501, 1637528780648]\) \(-3735772816268612449/909650165760000\) \(-343074119404584960000000000\) \([2]\) \(127401984\) \(3.8101\) \(\Gamma_0(N)\)-optimal
303450.cw6 303450cw2 \([1, 0, 1, -3932755501, 94923954380648]\) \(17836145204788591940449/770635366502400\) \(290644755199874510400000000\) \([2, 2]\) \(254803968\) \(4.1567\)  
303450.cw8 303450cw3 \([1, 0, 1, 1680780499, -11140081395352]\) \(1392333139184610040991/947901937500000000\) \(-357500756588124023437500000000\) \([2]\) \(382205952\) \(4.3594\)  
303450.cw3 303450cw4 \([1, 0, 1, -62923435501, 6075281111420648]\) \(73054578035931991395831649/136386452160\) \(51438084369956235000000\) \([2]\) \(509607936\) \(4.5032\)  
303450.cw5 303450cw5 \([1, 0, 1, -4129275501, 84912832540648]\) \(20645800966247918737249/3688936444974392640\) \(1391280593393501649704565000000\) \([2]\) \(509607936\) \(4.5032\)  
303450.cw4 303450cw6 \([1, 0, 1, -7350469501, -92909018895352]\) \(116454264690812369959009/57505157319440250000\) \(21688042228966313683628906250000\) \([2, 2]\) \(764411904\) \(4.7060\)  
303450.cw2 303450cw7 \([1, 0, 1, -63109407001, 6037563123479648]\) \(73704237235978088924479009/899277423164136103500\) \(339162044558852086306599867187500\) \([2]\) \(1528823808\) \(5.0525\)  
303450.cw1 303450cw8 \([1, 0, 1, -96091532001, -11456557036270352]\) \(260174968233082037895439009/223081361502731896500\) \(84135027435720856879212632812500\) \([2]\) \(1528823808\) \(5.0525\)