# Properties

 Label 52371e Number of curves 4 Conductor 52371 CM no Rank 2 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("52371.a1")

sage: E.isogeny_class()

## Elliptic curves in class 52371e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
52371.a3 52371e1 [1, -1, 1, -31046, -2080020] [2] 135168 $$\Gamma_0(N)$$-optimal
52371.a2 52371e2 [1, -1, 1, -54851, 1566906] [2, 2] 270336
52371.a4 52371e3 [1, -1, 1, 207004, 12041106] [2] 540672
52371.a1 52371e4 [1, -1, 1, -697586, 224210310] [2] 540672

## Rank

sage: E.rank()

The elliptic curves in class 52371e have rank $$2$$.

## Modular form 52371.2.a.a

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.