# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("p1")

sage: E.isogeny_class()

## Elliptic curves in class 1200p

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 $$[0, 1, 0, -2133408, 1198675188]$$ $$16778985534208729/81000$$ $$5184000000000$$ $$[4]$$ $$13824$$ $$2.0620$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1200.2.a.p

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.