Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1200.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1200.k1 1200p8 \([0, 1, 0, -2133408, 1198675188]\) \(16778985534208729/81000\) \(5184000000000\) \([4]\) \(13824\) \(2.0620\)  
1200.k2 1200p7 \([0, 1, 0, -181408, 3987188]\) \(10316097499609/5859375000\) \(375000000000000000\) \([2]\) \(13824\) \(2.0620\)  
1200.k3 1200p6 \([0, 1, 0, -133408, 18675188]\) \(4102915888729/9000000\) \(576000000000000\) \([2, 2]\) \(6912\) \(1.7155\)  
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.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.k8 1200p1 \([0, 1, 0, 592, -16812]\) \(357911/2160\) \(-138240000000\) \([2]\) \(1152\) \(0.81958\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 1200.2.a.k

sage: E.q_eigenform(10)
 
\(q + q^{3} - 4q^{7} + q^{9} - 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

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

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

The vertices are labelled with LMFDB labels.