↑

Properties

Label 3600bf
Number of curves $8$
Conductor $3600$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("bf1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 3600bf have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(11\) \( 1 - T + 11 T^{2}\) 1.11.ab
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 - 5 T + 17 T^{2}\) 1.17.af
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 - 2 T + 23 T^{2}\) 1.23.ac
\(29\) \( 1 - 8 T + 29 T^{2}\) 1.29.ai
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 3600bf do not have complex multiplication.

Modular form 3600.2.a.bf

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 3600bf

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3600.u7 3600bf1 \([0, 0, 0, -75, 40250]\) \(-1/15\) \(-699840000000\) \([2]\) \(3072\) \(0.95175\) \(\Gamma_0(N)\)-optimal
3600.u6 3600bf2 \([0, 0, 0, -18075, 922250]\) \(13997521/225\) \(10497600000000\) \([2, 2]\) \(6144\) \(1.2983\)  
3600.u5 3600bf3 \([0, 0, 0, -36075, -1219750]\) \(111284641/50625\) \(2361960000000000\) \([2, 2]\) \(12288\) \(1.6449\)  
3600.u4 3600bf4 \([0, 0, 0, -288075, 59512250]\) \(56667352321/15\) \(699840000000\) \([4]\) \(12288\) \(1.6449\)  
3600.u2 3600bf5 \([0, 0, 0, -486075, -130369750]\) \(272223782641/164025\) \(7652750400000000\) \([2, 2]\) \(24576\) \(1.9915\)  
3600.u8 3600bf6 \([0, 0, 0, 125925, -9157750]\) \(4733169839/3515625\) \(-164025000000000000\) \([2]\) \(24576\) \(1.9915\)  
3600.u1 3600bf7 \([0, 0, 0, -7776075, -8346199750]\) \(1114544804970241/405\) \(18895680000000\) \([2]\) \(49152\) \(2.3380\)  
3600.u3 3600bf8 \([0, 0, 0, -396075, -180139750]\) \(-147281603041/215233605\) \(-10041939074880000000\) \([2]\) \(49152\) \(2.3380\)