# Properties

 Label 52983.f Number of curves $6$ Conductor $52983$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("52983.f1")

sage: E.isogeny_class()

## Elliptic curves in class 52983.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
52983.f1 52983h4 [1, -1, 1, -1548799214, -23460303439834] [2] 10321920
52983.f2 52983h6 [1, -1, 1, -321448019, 1803354212756] [2] 20643840
52983.f3 52983h3 [1, -1, 1, -98654504, -351772016542] [2, 2] 10321920
52983.f4 52983h2 [1, -1, 1, -96800099, -366547915582] [2, 2] 5160960
52983.f5 52983h1 [1, -1, 1, -5934254, -5955896284] [4] 2580480 $$\Gamma_0(N)$$-optimal
52983.f6 52983h5 [1, -1, 1, 94468531, -1561262960140] [2] 20643840

## Rank

sage: E.rank()

The elliptic curves in class 52983.f have rank $$1$$.

## Modular form 52983.2.a.f

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + 2q^{5} + q^{7} + 3q^{8} - 2q^{10} + 4q^{11} - 2q^{13} - q^{14} - q^{16} + 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.