# Properties

 Label 33640.f Number of curves 4 Conductor 33640 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("33640.f1")

sage: E.isogeny_class()

## Elliptic curves in class 33640.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
33640.f1 33640g4 [0, 0, 0, -695507, -184039394]  430080
33640.f2 33640g2 [0, 0, 0, -207727, 33803154] [2, 2] 215040
33640.f3 33640g1 [0, 0, 0, -203522, 35339661]  107520 $$\Gamma_0(N)$$-optimal
33640.f4 33640g3 [0, 0, 0, 212773, 153309254]  430080

## Rank

sage: E.rank()

The elliptic curves in class 33640.f have rank $$1$$.

## Modular form 33640.2.a.f

sage: E.q_eigenform(10)

$$q + q^{5} - 3q^{9} - 2q^{13} + 6q^{17} + 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 