# Properties

 Label 435600.fs Number of curves $4$ Conductor $435600$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("fs1")

sage: E.isogeny_class()

## Elliptic curves in class 435600.fs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435600.fs1 435600fs4 $$[0, 0, 0, -6806150175, 216122709888250]$$ $$6749703004355978704/5671875$$ $$29300179546687500000000$$ $$[2]$$ $$159252480$$ $$4.0465$$ $$\Gamma_0(N)$$-optimal*
435600.fs2 435600fs3 $$[0, 0, 0, -425290800, 3378477466375]$$ $$-26348629355659264/24169921875$$ $$-7803669978698730468750000$$ $$[2]$$ $$79626240$$ $$3.7000$$ $$\Gamma_0(N)$$-optimal*
435600.fs3 435600fs2 $$[0, 0, 0, -85931175, 282320739250]$$ $$13584145739344/1195803675$$ $$6177368573899944300000000$$ $$[2]$$ $$53084160$$ $$3.4972$$ $$\Gamma_0(N)$$-optimal*
435600.fs4 435600fs1 $$[0, 0, 0, 5953200, 20542154875]$$ $$72268906496/606436875$$ $$-195798449820739218750000$$ $$[2]$$ $$26542080$$ $$3.1506$$ $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 435600.fs1.

## Rank

sage: E.rank()

The elliptic curves in class 435600.fs have rank $$0$$.

## Complex multiplication

The elliptic curves in class 435600.fs do not have complex multiplication.

## Modular form 435600.2.a.fs

sage: E.q_eigenform(10)

$$q - 2q^{7} + 2q^{13} + 2q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.