# Properties

 Label 1200a Number of curves $6$ Conductor $1200$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 1200a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1200.d5 1200a1 $$[0, -1, 0, 17, -38]$$ $$2048/3$$ $$-750000$$ $$[2]$$ $$128$$ $$-0.18721$$ $$\Gamma_0(N)$$-optimal
1200.d4 1200a2 $$[0, -1, 0, -108, -288]$$ $$35152/9$$ $$36000000$$ $$[2, 2]$$ $$256$$ $$0.15937$$
1200.d2 1200a3 $$[0, -1, 0, -1608, -24288]$$ $$28756228/3$$ $$48000000$$ $$[2]$$ $$512$$ $$0.50594$$
1200.d3 1200a4 $$[0, -1, 0, -608, 5712]$$ $$1556068/81$$ $$1296000000$$ $$[2, 2]$$ $$512$$ $$0.50594$$
1200.d1 1200a5 $$[0, -1, 0, -9608, 365712]$$ $$3065617154/9$$ $$288000000$$ $$[2]$$ $$1024$$ $$0.85251$$
1200.d6 1200a6 $$[0, -1, 0, 392, 21712]$$ $$207646/6561$$ $$-209952000000$$ $$[2]$$ $$1024$$ $$0.85251$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1200.2.a.a

sage: E.q_eigenform(10)

$$q - q^{3} + q^{9} - 4q^{11} + 2q^{13} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.