# Properties

 Label 338.f Number of curves $3$ Conductor $338$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("f1")

E.isogeny_class()

## Elliptic curves in class 338.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
338.f1 338c3 $$[1, 0, 0, -77659, -8336303]$$ $$-10730978619193/6656$$ $$-32127240704$$ $$[]$$ $$1008$$ $$1.3369$$
338.f2 338c2 $$[1, 0, 0, -764, -16264]$$ $$-10218313/17576$$ $$-84835994984$$ $$[]$$ $$336$$ $$0.78755$$
338.f3 338c1 $$[1, 0, 0, 81, 467]$$ $$12167/26$$ $$-125497034$$ $$[]$$ $$112$$ $$0.23825$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 338.f have rank $$0$$.

## Complex multiplication

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

## Modular form338.2.a.f

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.