# Properties

 Label 257600.fa Number of curves $2$ Conductor $257600$ CM no Rank $2$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("fa1")

sage: E.isogeny_class()

## Elliptic curves in class 257600.fa

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
257600.fa1 257600fa2 $$[0, -1, 0, -968833, 343953537]$$ $$24553362849625/1755162752$$ $$7189146632192000000$$ $$[2]$$ $$6193152$$ $$2.3652$$
257600.fa2 257600fa1 $$[0, -1, 0, 55167, 23441537]$$ $$4533086375/60669952$$ $$-248504123392000000$$ $$[2]$$ $$3096576$$ $$2.0186$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 257600.fa have rank $$2$$.

## Complex multiplication

The elliptic curves in class 257600.fa do not have complex multiplication.

## Modular form 257600.2.a.fa

sage: E.q_eigenform(10)

$$q + 2q^{3} - q^{7} + q^{9} - 4q^{11} - 6q^{17} + 6q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.