# Properties

 Label 435344.o Number of curves $3$ Conductor $435344$ CM no Rank $2$ Graph

# Learn more

Show commands: SageMath
sage: E = EllipticCurve("o1")

sage: E.isogeny_class()

## Elliptic curves in class 435344.o

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.o1 435344o3 $$[0, -1, 0, -40099398837, 3090699934894909]$$ $$-360675992659311050823073792/56219378022244619$$ $$-1111491378430658671373398016$$ $$[]$$ $$846526464$$ $$4.5939$$ $$\Gamma_0(N)$$-optimal*
435344.o2 435344o2 $$[0, -1, 0, -431421397, 5369381586269]$$ $$-449167881463536812032/369990050199923699$$ $$-7314928862066456615512027136$$ $$[]$$ $$282175488$$ $$4.0446$$ $$\Gamma_0(N)$$-optimal*
435344.o3 435344o1 $$[0, -1, 0, 43833643, -119695311971]$$ $$471114356703100928/585612268875179$$ $$-11577911582380580347129856$$ $$[]$$ $$94058496$$ $$3.4953$$ $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 435344.o1.

## Rank

sage: E.rank()

The elliptic curves in class 435344.o have rank $$2$$.

## Complex multiplication

The elliptic curves in class 435344.o do not have complex multiplication.

## Modular form 435344.2.a.o

sage: E.q_eigenform(10)

$$q - q^{3} - 3q^{5} + q^{7} - 2q^{9} - 3q^{11} + 3q^{15} - 6q^{17} - 7q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.