# Properties

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

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("u1")

sage: E.isogeny_class()

## Elliptic curves in class 435344.u

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.u1 435344u2 $$[0, -1, 0, -2615669, 1630276777]$$ $$-1601666731737088/1309814051$$ $$-1618488895921474304$$ $$[]$$ $$12192768$$ $$2.4230$$ $$\Gamma_0(N)$$-optimal*
435344.u2 435344u1 $$[0, -1, 0, 34251, 9718201]$$ $$3596091392/35177051$$ $$-43466984028226304$$ $$[]$$ $$4064256$$ $$1.8737$$ $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 435344.u1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 435344.2.a.u

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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.