# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("do1")

sage: E.isogeny_class()

## Elliptic curves in class 190575do

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
190575.dx6 190575do1 [1, -1, 0, 952308, -103907201909] [2] 22118400 $$\Gamma_0(N)$$-optimal
190575.dx5 190575do2 [1, -1, 0, -325883817, -2224093144784] [2, 2] 44236800
190575.dx4 190575do3 [1, -1, 0, -692740692, 3701012243341] [2] 88473600
190575.dx2 190575do4 [1, -1, 0, -5188404942, -143845020910409] [2, 2] 88473600
190575.dx3 190575do5 [1, -1, 0, -5162677317, -145342188592034] [2] 176947200
190575.dx1 190575do6 [1, -1, 0, -83014470567, -9206145580482284] [2] 176947200

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 190575do do not have complex multiplication.

## Modular form 190575.2.a.do

sage: E.q_eigenform(10)

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

## 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.