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Properties

Label 129360.bw
Number of curves $6$
Conductor $129360$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bw1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 129360.bw have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(7\)\(1\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 129360.bw do not have complex multiplication.

Modular form 129360.2.a.bw

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} + q^{11} + 2 q^{13} + q^{15} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 129360.bw

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
129360.bw1 129360ef4 \([0, -1, 0, -69941369136, 7119520258078656]\) \(78519570041710065450485106721/96428056919040\) \(46467745662845488988160\) \([2]\) \(212336640\) \(4.5235\)  
129360.bw2 129360ef6 \([0, -1, 0, -20571070256, -1039409537294400]\) \(1997773216431678333214187041/187585177195046990066400\) \(90395478064415061337382476185600\) \([2]\) \(424673280\) \(4.8701\)  
129360.bw3 129360ef3 \([0, -1, 0, -4568062256, 100683159844800]\) \(21876183941534093095979041/3572502915711058560000\) \(1721554516092888385640202240000\) \([2, 2]\) \(212336640\) \(4.5235\)  
129360.bw4 129360ef2 \([0, -1, 0, -4371372336, 111241632102336]\) \(19170300594578891358373921/671785075055001600\) \(323726714040917537744486400\) \([2, 2]\) \(106168320\) \(4.1770\)  
129360.bw5 129360ef1 \([0, -1, 0, -260954416, 1901227096000]\) \(-4078208988807294650401/880065599546327040\) \(-424095079305321799389020160\) \([2]\) \(53084160\) \(3.8304\) \(\Gamma_0(N)\)-optimal
129360.bw6 129360ef5 \([0, -1, 0, 8287907024, 565030485462976]\) \(130650216943167617311657439/361816948816603087500000\) \(-174356079457585302082713600000000\) \([2]\) \(424673280\) \(4.8701\)