# Properties

 Label 458640gr Number of curves $2$ Conductor $458640$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("gr1")

sage: E.isogeny_class()

## Elliptic curves in class 458640gr

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
458640.gr2 458640gr1 $$[0, 0, 0, 28077, 5026322]$$ $$6967871/35100$$ $$-12330560909721600$$ $$$$ $$3317760$$ $$1.7698$$ $$\Gamma_0(N)$$-optimal*
458640.gr1 458640gr2 $$[0, 0, 0, -324723, 63802802]$$ $$10779215329/1232010$$ $$432802687931228160$$ $$$$ $$6635520$$ $$2.1164$$ $$\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 458640gr1.

## Rank

sage: E.rank()

The elliptic curves in class 458640gr have rank $$1$$.

## Complex multiplication

The elliptic curves in class 458640gr do not have complex multiplication.

## Modular form 458640.2.a.gr

sage: E.q_eigenform(10)

$$q - q^{5} + 4q^{11} + q^{13} + 8q^{17} - 6q^{19} + O(q^{20})$$ ## 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. 