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("1200.k1")
sage: E.isogeny_class()

Elliptic curves in class 1200.k

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
1200.k1 1200p8 [0, 1, 0, -2133408, 1198675188] 4 13824  
1200.k2 1200p7 [0, 1, 0, -181408, 3987188] 2 13824  
1200.k3 1200p6 [0, 1, 0, -133408, 18675188] 4 6912  
1200.k4 1200p4 [0, 1, 0, -115408, -15128812] 2 4608  
1200.k5 1200p5 [0, 1, 0, -27408, 1495188] 4 4608  
1200.k6 1200p2 [0, 1, 0, -7408, -224812] 4 2304  
1200.k7 1200p3 [0, 1, 0, -5408, 499188] 2 3456  
1200.k8 1200p1 [0, 1, 0, 592, -16812] 2 1152 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()

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

Modular form 1200.2.a.k

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

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.