# Properties

 Label 388080fd Number of curves $4$ Conductor $388080$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("fd1")

sage: E.isogeny_class()

## Elliptic curves in class 388080fd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
388080.fd4 388080fd1 $$[0, 0, 0, 96432, 42349867]$$ $$72268906496/606436875$$ $$-832187814401790000$$ $$[2]$$ $$3317760$$ $$2.1199$$ $$\Gamma_0(N)$$-optimal
388080.fd3 388080fd2 $$[0, 0, 0, -1391943, 582034642]$$ $$13584145739344/1195803675$$ $$26255217326667436800$$ $$[2]$$ $$6635520$$ $$2.4665$$
388080.fd2 388080fd3 $$[0, 0, 0, -6889008, 6965095543]$$ $$-26348629355659264/24169921875$$ $$-33167367105468750000$$ $$[2]$$ $$9953280$$ $$2.6692$$
388080.fd1 388080fd4 $$[0, 0, 0, -110248383, 445560267418]$$ $$6749703004355978704/5671875$$ $$124532407692000000$$ $$[2]$$ $$19906560$$ $$3.0158$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 388080.2.a.fd

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.