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Properties

Label 1200.e
Number of curves $8$
Conductor $1200$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("e1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 1200.e have rank \(0\).

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 \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 1200.2.a.e

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 1200.e

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1200.e1 1200j7 \([0, -1, 0, -864008, 309406512]\) \(1114544804970241/405\) \(25920000000\) \([4]\) \(6144\) \(1.7887\)  
1200.e2 1200j5 \([0, -1, 0, -54008, 4846512]\) \(272223782641/164025\) \(10497600000000\) \([2, 2]\) \(3072\) \(1.4422\)  
1200.e3 1200j8 \([0, -1, 0, -44008, 6686512]\) \(-147281603041/215233605\) \(-13774950720000000\) \([2]\) \(6144\) \(1.7887\)  
1200.e4 1200j3 \([0, -1, 0, -32008, -2193488]\) \(56667352321/15\) \(960000000\) \([2]\) \(1536\) \(1.0956\)  
1200.e5 1200j4 \([0, -1, 0, -4008, 46512]\) \(111284641/50625\) \(3240000000000\) \([2, 2]\) \(1536\) \(1.0956\)  
1200.e6 1200j2 \([0, -1, 0, -2008, -33488]\) \(13997521/225\) \(14400000000\) \([2, 2]\) \(768\) \(0.74901\)  
1200.e7 1200j1 \([0, -1, 0, -8, -1488]\) \(-1/15\) \(-960000000\) \([2]\) \(384\) \(0.40244\) \(\Gamma_0(N)\)-optimal
1200.e8 1200j6 \([0, -1, 0, 13992, 334512]\) \(4733169839/3515625\) \(-225000000000000\) \([2]\) \(3072\) \(1.4422\)