# Properties

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

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("bc1")

sage: E.isogeny_class()

## Elliptic curves in class 435344bc

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.bc3 435344bc1 $$[0, 0, 0, -1532999, -730568410]$$ $$322440248841552/27209$$ $$33621157396736$$ $$[2]$$ $$3010560$$ $$2.0376$$ $$\Gamma_0(N)$$-optimal*
435344.bc2 435344bc2 $$[0, 0, 0, -1536379, -727185030]$$ $$81144432781668/740329681$$ $$3659192286431159296$$ $$[2, 2]$$ $$6021120$$ $$2.3841$$ $$\Gamma_0(N)$$-optimal*
435344.bc1 435344bc3 $$[0, 0, 0, -2678819, 499567042]$$ $$215062038362754/113550802729$$ $$1122484298894462486528$$ $$[2]$$ $$12042240$$ $$2.7307$$ $$\Gamma_0(N)$$-optimal*
435344.bc4 435344bc4 $$[0, 0, 0, -448019, -1737400782]$$ $$-1006057824354/131332646081$$ $$-1298263240903855163392$$ $$[2]$$ $$12042240$$ $$2.7307$$
*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 435344bc1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 435344.2.a.bc

sage: E.q_eigenform(10)

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

## 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 Cremona 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 Cremona labels.