# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("1200.k1")

sage: E.isogeny_class()

## Elliptic curves in class 1200p

sage: E.isogeny_class().curves

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

## Rank

sage: E.rank()

The elliptic curves in class 1200p have rank $$1$$.

## Modular form1200.2.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 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

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

The vertices are labelled with Cremona labels.