# Properties

 Label 388416fe Number of curves $4$ Conductor $388416$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("fe1")

sage: E.isogeny_class()

## Elliptic curves in class 388416fe

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
388416.fe4 388416fe1 $$[0, 1, 0, 18111, -253475649]$$ $$103823/4386816$$ $$-27757661097899851776$$ $$[2]$$ $$10616832$$ $$2.4097$$ $$\Gamma_0(N)$$-optimal
388416.fe3 388416fe2 $$[0, 1, 0, -5900609, -5420518209]$$ $$3590714269297/73410624$$ $$464507109935167832064$$ $$[2, 2]$$ $$21233664$$ $$2.7563$$
388416.fe2 388416fe3 $$[0, 1, 0, -12559169, 9051196095]$$ $$34623662831857/14438442312$$ $$91359516441554328748032$$ $$[2]$$ $$42467328$$ $$3.1029$$
388416.fe1 388416fe4 $$[0, 1, 0, -93941569, -350488256833]$$ $$14489843500598257/6246072$$ $$39522138555408187392$$ $$[2]$$ $$42467328$$ $$3.1029$$

## Rank

sage: E.rank()

The elliptic curves in class 388416fe have rank $$1$$.

## Complex multiplication

The elliptic curves in class 388416fe do not have complex multiplication.

## Modular form 388416.2.a.fe

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{5} - q^{7} + q^{9} + 6q^{13} - 2q^{15} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.