# Properties

 Label 58989a Number of curves 6 Conductor 58989 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("58989.n1")

sage: E.isogeny_class()

## Elliptic curves in class 58989a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
58989.n6 58989a1 [1, 1, 0, 2751, -11088] [2] 74880 $$\Gamma_0(N)$$-optimal
58989.n5 58989a2 [1, 1, 0, -11294, -103785] [2, 2] 149760
58989.n3 58989a3 [1, 1, 0, -109609, 13837282] [2] 299520
58989.n2 58989a4 [1, 1, 0, -137699, -19696560] [2, 2] 299520
58989.n4 58989a5 [1, 1, 0, -95564, -31924137] [2] 599040
58989.n1 58989a6 [1, 1, 0, -2202314, -1258878483] [2] 599040

## Rank

sage: E.rank()

The elliptic curves in class 58989a have rank $$1$$.

## Modular form 58989.2.a.n

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} - q^{7} - 3q^{8} + q^{9} + 2q^{10} + 4q^{11} + q^{12} - 2q^{13} - q^{14} - 2q^{15} - q^{16} - 6q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.