# Properties

 Label 258570eg Number of curves 4 Conductor 258570 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("258570.eg1")

sage: E.isogeny_class()

## Elliptic curves in class 258570eg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
258570.eg4 258570eg1 [1, -1, 1, 2403148, -243740649] [4] 15482880 $$\Gamma_0(N)$$-optimal
258570.eg3 258570eg2 [1, -1, 1, -9764852, -1956995049] [2, 2] 30965760
258570.eg2 258570eg3 [1, -1, 1, -97678652, 369672220311] [2] 61931520
258570.eg1 258570eg4 [1, -1, 1, -116539052, -483380508009] [2] 61931520

## Rank

sage: E.rank()

The elliptic curves in class 258570eg have rank $$1$$.

## Modular form 258570.2.a.eg

sage: E.q_eigenform(10)

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

## 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 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.