# Properties

 Label 10.0.5586055083.1 Degree $10$ Signature $[0, 5]$ Discriminant $-5586055083$ Root discriminant $$9.43$$ Ramified primes $3,7,40543$ Class number $1$ Class group trivial Galois group $S_5^2 \wr C_2$ (as 10T43)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 3*x^8 - 3*x^7 + 6*x^6 + 6*x^4 + 3*x^3 + 3*x^2 + 2*x + 1)

gp: K = bnfinit(y^10 - y^9 + 3*y^8 - 3*y^7 + 6*y^6 + 6*y^4 + 3*y^3 + 3*y^2 + 2*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - x^9 + 3*x^8 - 3*x^7 + 6*x^6 + 6*x^4 + 3*x^3 + 3*x^2 + 2*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - x^9 + 3*x^8 - 3*x^7 + 6*x^6 + 6*x^4 + 3*x^3 + 3*x^2 + 2*x + 1)

$$x^{10} - x^{9} + 3x^{8} - 3x^{7} + 6x^{6} + 6x^{4} + 3x^{3} + 3x^{2} + 2x + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

## Invariants

 Degree: $10$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K);  oscar: degree(K) Signature: $[0, 5]$ sage: K.signature()  gp: K.sign  magma: Signature(K);  oscar: signature(K) Discriminant: $$-5586055083$$ -5586055083 $$\medspace = -\,3^{9}\cdot 7\cdot 40543$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$9.43$$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(OK))^(1/Degree(K));  oscar: (1.0 * dK)^(1/degree(K)) Galois root discriminant: $3^{9/10}7^{1/2}40543^{1/2}\approx 1431.9112129325747$ Ramified primes: $$3$$, $$7$$, $$40543$$ 3, 7, 40543 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q(\sqrt{-851403})$$ $\card{ \Aut(K/\Q) }$: $1$ sage: K.automorphisms()  magma: Automorphisms(K);  oscar: automorphisms(K) This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{23}a^{9}+\frac{6}{23}a^{8}-\frac{1}{23}a^{7}-\frac{10}{23}a^{6}+\frac{5}{23}a^{5}-\frac{11}{23}a^{4}-\frac{2}{23}a^{3}-\frac{11}{23}a^{2}-\frac{5}{23}a-\frac{10}{23}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

 Monogenic: Not computed Index: $1$ Inessential primes: None

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

## Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

 Rank: $4$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K);  oscar: rank(UK) Torsion generator: $$-\frac{2}{23} a^{9} + \frac{11}{23} a^{8} - \frac{21}{23} a^{7} + \frac{43}{23} a^{6} - \frac{56}{23} a^{5} + \frac{68}{23} a^{4} - \frac{42}{23} a^{3} + \frac{45}{23} a^{2} + \frac{10}{23} a + \frac{20}{23}$$ -(2)/(23)*a^(9) + (11)/(23)*a^(8) - (21)/(23)*a^(7) + (43)/(23)*a^(6) - (56)/(23)*a^(5) + (68)/(23)*a^(4) - (42)/(23)*a^(3) + (45)/(23)*a^(2) + (10)/(23)*a + (20)/(23)  (order $6$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);  oscar: torsion_units_generator(OK) Fundamental units: $\frac{9}{23}a^{9}-\frac{15}{23}a^{8}+\frac{37}{23}a^{7}-\frac{44}{23}a^{6}+\frac{68}{23}a^{5}-\frac{30}{23}a^{4}+\frac{51}{23}a^{3}+\frac{16}{23}a^{2}+\frac{1}{23}a+\frac{2}{23}$, $\frac{12}{23}a^{9}-\frac{20}{23}a^{8}+\frac{57}{23}a^{7}-\frac{74}{23}a^{6}+\frac{129}{23}a^{5}-\frac{86}{23}a^{4}+\frac{137}{23}a^{3}-\frac{17}{23}a^{2}+\frac{55}{23}a+\frac{18}{23}$, $\frac{4}{23}a^{9}+\frac{1}{23}a^{8}-\frac{4}{23}a^{7}+\frac{6}{23}a^{6}-\frac{3}{23}a^{5}+\frac{25}{23}a^{4}+\frac{15}{23}a^{3}-\frac{21}{23}a^{2}+\frac{26}{23}a-\frac{17}{23}$, $\frac{8}{23}a^{9}+\frac{2}{23}a^{8}+\frac{15}{23}a^{7}-\frac{11}{23}a^{6}+\frac{40}{23}a^{5}+\frac{27}{23}a^{4}+\frac{76}{23}a^{3}+\frac{27}{23}a^{2}+\frac{29}{23}a+\frac{35}{23}$ 9/23*a^9 - 15/23*a^8 + 37/23*a^7 - 44/23*a^6 + 68/23*a^5 - 30/23*a^4 + 51/23*a^3 + 16/23*a^2 + 1/23*a + 2/23, 12/23*a^9 - 20/23*a^8 + 57/23*a^7 - 74/23*a^6 + 129/23*a^5 - 86/23*a^4 + 137/23*a^3 - 17/23*a^2 + 55/23*a + 18/23, 4/23*a^9 + 1/23*a^8 - 4/23*a^7 + 6/23*a^6 - 3/23*a^5 + 25/23*a^4 + 15/23*a^3 - 21/23*a^2 + 26/23*a - 17/23, 8/23*a^9 + 2/23*a^8 + 15/23*a^7 - 11/23*a^6 + 40/23*a^5 + 27/23*a^4 + 76/23*a^3 + 27/23*a^2 + 29/23*a + 35/23 sage: UK.fundamental_units()  gp: K.fu  magma: [K|fUK(g): g in Generators(UK)];  oscar: [K(fUK(a)) for a in gens(UK)] Regulator: $$17.1897439668$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);  oscar: regulator(K)

## Class number formula

\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 17.1897439668 \cdot 1}{6\cdot\sqrt{5586055083}}\cr\approx \mathstrut & 0.375374608788 \end{aligned}

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 3*x^8 - 3*x^7 + 6*x^6 + 6*x^4 + 3*x^3 + 3*x^2 + 2*x + 1)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^10 - x^9 + 3*x^8 - 3*x^7 + 6*x^6 + 6*x^4 + 3*x^3 + 3*x^2 + 2*x + 1, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^10 - x^9 + 3*x^8 - 3*x^7 + 6*x^6 + 6*x^4 + 3*x^3 + 3*x^2 + 2*x + 1);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - x^9 + 3*x^8 - 3*x^7 + 6*x^6 + 6*x^4 + 3*x^3 + 3*x^2 + 2*x + 1);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

## Galois group

$S_5\wr C_2$ (as 10T43):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

 A non-solvable group of order 28800 The 35 conjugacy class representatives for $S_5^2 \wr C_2$ Character table for $S_5^2 \wr C_2$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

## Sibling fields

 Degree 12 sibling: data not computed Degree 20 siblings: data not computed Degree 24 siblings: data not computed Degree 25 sibling: data not computed Degree 30 sibling: data not computed Degree 36 sibling: data not computed Degree 40 siblings: data not computed Minimal sibling: This field is its own minimal sibling

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.10.0.1}{10} }$ R ${\href{/padicField/5.10.0.1}{10} }$ R ${\href{/padicField/11.10.0.1}{10} }$ ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])