# Properties

 Label 259182fh Number of curves $6$ Conductor $259182$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("259182.fh1")

sage: E.isogeny_class()

## Elliptic curves in class 259182fh

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
259182.fh5 259182fh1 [1, -1, 1, -76495739, 257212812651] [2] 39321600 $$\Gamma_0(N)$$-optimal
259182.fh4 259182fh2 [1, -1, 1, -98798459, 95018511723] [2, 2] 78643200
259182.fh6 259182fh3 [1, -1, 1, 381058501, 747048148971] [2] 157286400
259182.fh2 259182fh4 [1, -1, 1, -935498939, -10937714017557] [2, 2] 157286400
259182.fh3 259182fh5 [1, -1, 1, -318645779, -25146803187525] [2] 314572800
259182.fh1 259182fh6 [1, -1, 1, -14939559779, -702833547127269] [2] 314572800

## Rank

sage: E.rank()

The elliptic curves in class 259182fh have rank $$1$$.

## Modular form 259182.2.a.fh

sage: E.q_eigenform(10)

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

## 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 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.