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Properties

Label 129360.cb
Number of curves $6$
Conductor $129360$
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 129360.cb have rank \(1\).

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
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 129360.2.a.cb

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} + q^{11} + 2 q^{13} + q^{15} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.cb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
129360.cb1 129360y4 \([0, -1, 0, -2587216, 1602621616]\) \(15897679904620804/2475\) \(298169625600\) \([2]\) \(1572864\) \(2.0483\)  
129360.cb2 129360y6 \([0, -1, 0, -1372016, -606255264]\) \(1185450336504002/26043266205\) \(6274998734340188160\) \([2]\) \(3145728\) \(2.3949\)  
129360.cb3 129360y3 \([0, -1, 0, -186216, 17001216]\) \(5927735656804/2401490025\) \(289313689550054400\) \([2, 2]\) \(1572864\) \(2.0483\)  
129360.cb4 129360y2 \([0, -1, 0, -161716, 25076416]\) \(15529488955216/6125625\) \(184492455840000\) \([2, 2]\) \(786432\) \(1.7018\)  
129360.cb5 129360y1 \([0, -1, 0, -8591, 515166]\) \(-37256083456/38671875\) \(-72795318750000\) \([2]\) \(393216\) \(1.3552\) \(\Gamma_0(N)\)-optimal
129360.cb6 129360y5 \([0, -1, 0, 607584, 123052896]\) \(102949393183198/86815346805\) \(-20917736931863439360\) \([2]\) \(3145728\) \(2.3949\)