# Properties

 Label 457776er Number of curves $2$ Conductor $457776$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("er1")

sage: E.isogeny_class()

## Elliptic curves in class 457776er

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
457776.er2 457776er1 [0, 0, 0, -500259, 110375458] [2] 5308416 $$\Gamma_0(N)$$-optimal*
457776.er1 457776er2 [0, 0, 0, -7574979, 8024157250] [2] 10616832 $$\Gamma_0(N)$$-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 457776er1.

## Rank

sage: E.rank()

The elliptic curves in class 457776er have rank $$0$$.

## Complex multiplication

The elliptic curves in class 457776er do not have complex multiplication.

## Modular form 457776.2.a.er

sage: E.q_eigenform(10)

$$q + 2q^{5} - 2q^{7} + q^{11} + 4q^{13} - 2q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.