# Properties

 Label 34969f Number of curves $2$ Conductor $34969$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 34969f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
34969.k2 34969f1 [1, 0, 1, -729, -29647] [] 28704 $$\Gamma_0(N)$$-optimal
34969.k1 34969f2 [1, 0, 1, -1049799, 413933375] [] 315744

## Rank

sage: E.rank()

The elliptic curves in class 34969f have rank $$0$$.

## Complex multiplication

The elliptic curves in class 34969f do not have complex multiplication.

## Modular form 34969.2.a.f

sage: E.q_eigenform(10)

$$q + q^{2} - 2q^{3} - q^{4} - q^{5} - 2q^{6} + 2q^{7} - 3q^{8} + q^{9} - q^{10} + 2q^{12} + q^{13} + 2q^{14} + 2q^{15} - q^{16} + q^{18} + 6q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 