# Properties

 Label 38720.f Number of curves 4 Conductor 38720 CM no Rank 1 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("38720.f1")

sage: E.isogeny_class()

## Elliptic curves in class 38720.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
38720.f1 38720cj4 [0, 1, 0, -3436561, -2453222961] [2] 829440
38720.f2 38720cj3 [0, 1, 0, -215541, -38102165] [2] 414720
38720.f3 38720cj2 [0, 1, 0, -48561, -2343761] [2] 276480
38720.f4 38720cj1 [0, 1, 0, -21941, 1217995] [2] 138240 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 38720.f have rank $$1$$.

## Modular form 38720.2.a.f

sage: E.q_eigenform(10)

$$q - 2q^{3} - q^{5} - 4q^{7} + q^{9} - 4q^{13} + 2q^{15} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.