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Properties

Label 1200p
Number of curves $8$
Conductor $1200$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("p1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 1200p have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(11\) \( 1 + 2 T + 11 T^{2}\) 1.11.c
\(13\) \( 1 - T + 13 T^{2}\) 1.13.ab
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 5 T + 19 T^{2}\) 1.19.af
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 1200p do not have complex multiplication.

Modular form 1200.2.a.p

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - 4 q^{7} + q^{9} - 2 q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 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 1200p

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1200.k8 1200p1 \([0, 1, 0, 592, -16812]\) \(357911/2160\) \(-138240000000\) \([2]\) \(1152\) \(0.81958\) \(\Gamma_0(N)\)-optimal
1200.k6 1200p2 \([0, 1, 0, -7408, -224812]\) \(702595369/72900\) \(4665600000000\) \([2, 2]\) \(2304\) \(1.1662\)  
1200.k7 1200p3 \([0, 1, 0, -5408, 499188]\) \(-273359449/1536000\) \(-98304000000000\) \([2]\) \(3456\) \(1.3689\)  
1200.k4 1200p4 \([0, 1, 0, -115408, -15128812]\) \(2656166199049/33750\) \(2160000000000\) \([2]\) \(4608\) \(1.5127\)  
1200.k5 1200p5 \([0, 1, 0, -27408, 1495188]\) \(35578826569/5314410\) \(340122240000000\) \([4]\) \(4608\) \(1.5127\)  
1200.k3 1200p6 \([0, 1, 0, -133408, 18675188]\) \(4102915888729/9000000\) \(576000000000000\) \([2, 2]\) \(6912\) \(1.7155\)  
1200.k2 1200p7 \([0, 1, 0, -181408, 3987188]\) \(10316097499609/5859375000\) \(375000000000000000\) \([2]\) \(13824\) \(2.0620\)  
1200.k1 1200p8 \([0, 1, 0, -2133408, 1198675188]\) \(16778985534208729/81000\) \(5184000000000\) \([4]\) \(13824\) \(2.0620\)