# Properties

 Label 38088i Number of curves $6$ Conductor $38088$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 38088i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38088.g5 38088i1 $$[0, 0, 0, 3174, 85169]$$ $$2048/3$$ $$-5180071827888$$ $$[2]$$ $$45056$$ $$1.1251$$ $$\Gamma_0(N)$$-optimal
38088.g4 38088i2 $$[0, 0, 0, -20631, 851690]$$ $$35152/9$$ $$248643447738624$$ $$[2, 2]$$ $$90112$$ $$1.4717$$
38088.g3 38088i3 $$[0, 0, 0, -115851, -14478730]$$ $$1556068/81$$ $$8951164118590464$$ $$[2, 2]$$ $$180224$$ $$1.8183$$
38088.g2 38088i4 $$[0, 0, 0, -306291, 65239454]$$ $$28756228/3$$ $$331524596984832$$ $$[2]$$ $$180224$$ $$1.8183$$
38088.g6 38088i5 $$[0, 0, 0, 74589, -57403906]$$ $$207646/6561$$ $$-1450088587211655168$$ $$[2]$$ $$360448$$ $$2.1648$$
38088.g1 38088i6 $$[0, 0, 0, -1829811, -952700434]$$ $$3065617154/9$$ $$1989147581908992$$ $$[2]$$ $$360448$$ $$2.1648$$

## Rank

sage: E.rank()

The elliptic curves in class 38088i have rank $$1$$.

## Complex multiplication

The elliptic curves in class 38088i do not have complex multiplication.

## Modular form 38088.2.a.i

sage: E.q_eigenform(10)

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

## 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.