# Properties

 Label 388080hk Number of curves $2$ Conductor $388080$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("hk1")

sage: E.isogeny_class()

## Elliptic curves in class 388080hk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
388080.hk2 388080hk1 $$[0, 0, 0, -9408, -1101373]$$ $$-67108864/343035$$ $$-470732501075760$$ $$$$ $$1474560$$ $$1.4998$$ $$\Gamma_0(N)$$-optimal
388080.hk1 388080hk2 $$[0, 0, 0, -227703, -41747902]$$ $$59466754384/121275$$ $$2662729299014400$$ $$$$ $$2949120$$ $$1.8463$$

## Rank

sage: E.rank()

The elliptic curves in class 388080hk have rank $$1$$.

## Complex multiplication

The elliptic curves in class 388080hk do not have complex multiplication.

## Modular form 388080.2.a.hk

sage: E.q_eigenform(10)

$$q - q^{5} + q^{11} + 6q^{13} + 2q^{17} + O(q^{20})$$ ## 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. 