# Properties

 Label 338.a Number of curves $2$ Conductor $338$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("a1")

E.isogeny_class()

## Elliptic curves in class 338.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
338.a1 338f2 $$[1, -1, 0, -35944, -2868878]$$ $$-1064019559329/125497034$$ $$-605750213184506$$ $$[]$$ $$2352$$ $$1.5723$$
338.a2 338f1 $$[1, -1, 0, -454, 5812]$$ $$-2146689/1664$$ $$-8031810176$$ $$[]$$ $$336$$ $$0.59931$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 338.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 338.a do not have complex multiplication.

## Modular form338.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - 3 q^{3} + q^{4} + q^{5} + 3 q^{6} - q^{7} - q^{8} + 6 q^{9} - q^{10} + 2 q^{11} - 3 q^{12} + q^{14} - 3 q^{15} + q^{16} - 3 q^{17} - 6 q^{18} - 6 q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.