# Properties

 Label 123200fs Number of curves 4 Conductor 123200 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("123200.ha1")

sage: E.isogeny_class()

## Elliptic curves in class 123200fs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
123200.ha3 123200fs1 [0, -1, 0, -89633, -200972863] [2] 2654208 $$\Gamma_0(N)$$-optimal
123200.ha2 123200fs2 [0, -1, 0, -5721633, -5219084863] [2] 5308416
123200.ha4 123200fs3 [0, -1, 0, 806367, 5414259137] [2] 7962624
123200.ha1 123200fs4 [0, -1, 0, -41785633, 101033299137] [2] 15925248

## Rank

sage: E.rank()

The elliptic curves in class 123200fs have rank $$1$$.

## Modular form 123200.2.a.ha

sage: E.q_eigenform(10)

$$q + 2q^{3} + q^{7} + q^{9} - q^{11} - 4q^{13} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.