# Properties

 Label 58800gk Number of curves $2$ Conductor $58800$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("gk1")

sage: E.isogeny_class()

## Elliptic curves in class 58800gk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
58800.f1 58800gk1 $$[0, -1, 0, -427408, 61477312]$$ $$393349474783/153600000$$ $$3371827200000000000$$ $$[2]$$ $$1290240$$ $$2.2539$$ $$\Gamma_0(N)$$-optimal
58800.f2 58800gk2 $$[0, -1, 0, 1364592, 441381312]$$ $$12801408679457/11250000000$$ $$-246960000000000000000$$ $$[2]$$ $$2580480$$ $$2.6005$$

## Rank

sage: E.rank()

The elliptic curves in class 58800gk have rank $$1$$.

## Complex multiplication

The elliptic curves in class 58800gk do not have complex multiplication.

## Modular form 58800.2.a.gk

sage: E.q_eigenform(10)

$$q - q^{3} + q^{9} - 6q^{11} + 6q^{13} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.