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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("f1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 3600.f 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 \(\mathbb{F}_p\)
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 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 3600.f do not have complex multiplication.

Modular form 3600.2.a.f

Copy content sage:E.q_eigenform(10)
 
\(q - 4 q^{7} - 2 q^{13} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 3600.f

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3600.f1 3600bk7 \([0, 0, 0, -19200675, -32383430750]\) \(16778985534208729/81000\) \(3779136000000000\) \([2]\) \(110592\) \(2.6113\)  
3600.f2 3600bk8 \([0, 0, 0, -1632675, -109286750]\) \(10316097499609/5859375000\) \(273375000000000000000\) \([2]\) \(110592\) \(2.6113\)  
3600.f3 3600bk6 \([0, 0, 0, -1200675, -505430750]\) \(4102915888729/9000000\) \(419904000000000000\) \([2, 2]\) \(55296\) \(2.2648\)  
3600.f4 3600bk5 \([0, 0, 0, -1038675, 407439250]\) \(2656166199049/33750\) \(1574640000000000\) \([2]\) \(36864\) \(2.0620\)  
3600.f5 3600bk4 \([0, 0, 0, -246675, -40616750]\) \(35578826569/5314410\) \(247949112960000000\) \([2]\) \(36864\) \(2.0620\)  
3600.f6 3600bk2 \([0, 0, 0, -66675, 6003250]\) \(702595369/72900\) \(3401222400000000\) \([2, 2]\) \(18432\) \(1.7155\)  
3600.f7 3600bk3 \([0, 0, 0, -48675, -13526750]\) \(-273359449/1536000\) \(-71663616000000000\) \([2]\) \(27648\) \(1.9182\)  
3600.f8 3600bk1 \([0, 0, 0, 5325, 459250]\) \(357911/2160\) \(-100776960000000\) \([2]\) \(9216\) \(1.3689\) \(\Gamma_0(N)\)-optimal