# Properties

 Label 190575ds Number of curves 6 Conductor 190575 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("190575.el1")

sage: E.isogeny_class()

## Elliptic curves in class 190575ds

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
190575.el4 190575ds1 [1, -1, 0, -926217, 343218816] [2] 2457600 $$\Gamma_0(N)$$-optimal
190575.el3 190575ds2 [1, -1, 0, -1062342, 235816191] [2, 2] 4915200
190575.el6 190575ds3 [1, -1, 0, 3429783, 1704741066] [2] 9830400
190575.el2 190575ds4 [1, -1, 0, -7732467, -8108510184] [2, 2] 9830400
190575.el5 190575ds5 [1, -1, 0, 843408, -25114470309] [2] 19660800
190575.el1 190575ds6 [1, -1, 0, -123030342, -525219479559] [2] 19660800

## Rank

sage: E.rank()

The elliptic curves in class 190575ds have rank $$0$$.

## Modular form 190575.2.a.el

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} + q^{7} - 3q^{8} + 6q^{13} + q^{14} - q^{16} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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 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.