# Properties

 Label 388080km Number of curves $6$ Conductor $388080$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("km1")

sage: E.isogeny_class()

## Elliptic curves in class 388080km

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
388080.km6 388080km1 [0, 0, 0, 246813, 13709786146] [2] 23592960 $$\Gamma_0(N)$$-optimal
388080.km5 388080km2 [0, 0, 0, -84460467, 293464049074] [2, 2] 47185920
388080.km2 388080km3 [0, 0, 0, -1344697347, 18979500363586] [2, 2] 94371840
388080.km4 388080km4 [0, 0, 0, -179540067, -488299438046] [2] 94371840
388080.km1 388080km5 [0, 0, 0, -21515155347, 1214688284695186] [4] 188743680
388080.km3 388080km6 [0, 0, 0, -1338029427, 19177040160754] [2] 188743680

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 388080.2.a.km

sage: E.q_eigenform(10)

$$q + q^{5} - q^{11} + 2q^{13} + 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.