# Properties

 Label 435344.z Number of curves $4$ Conductor $435344$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 435344.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.z1 435344z4 $$[0, 0, 0, -334451, -74429966]$$ $$209267191953/55223$$ $$1091792377475072$$ $$[2]$$ $$2457600$$ $$1.8700$$
435344.z2 435344z2 $$[0, 0, 0, -23491, -856830]$$ $$72511713/25921$$ $$512473973100544$$ $$[2, 2]$$ $$1228800$$ $$1.5234$$
435344.z3 435344z1 $$[0, 0, 0, -9971, 373490]$$ $$5545233/161$$ $$3183068155904$$ $$[2]$$ $$614400$$ $$1.1768$$ $$\Gamma_0(N)$$-optimal*
435344.z4 435344z3 $$[0, 0, 0, 71149, -6024174]$$ $$2014698447/1958887$$ $$-38728390252883968$$ $$[2]$$ $$2457600$$ $$1.8700$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 435344.z1.

## Rank

sage: E.rank()

The elliptic curves in class 435344.z have rank $$0$$.

## Complex multiplication

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

## Modular form 435344.2.a.z

sage: E.q_eigenform(10)

$$q - 2q^{5} + q^{7} - 3q^{9} + 4q^{11} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.