↑

Properties

Label 10.2.299...096.1
Degree $10$
Signature $[2, 4]$
Discriminant $2.994\times 10^{18}$
Root discriminant \(70.41\)
Ramified primes $2,61$
Class number $5$ (GRH)
Class group [5] (GRH)
Galois group $D_{10}$ (as 10T3)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 - 27*x^8 + 56*x^7 + 222*x^6 - 622*x^5 - 1890*x^4 + 2792*x^3 + 11345*x^2 - 225*x + 2061)
 
gp: K = bnfinit(y^10 - y^9 - 27*y^8 + 56*y^7 + 222*y^6 - 622*y^5 - 1890*y^4 + 2792*y^3 + 11345*y^2 - 225*y + 2061, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - x^9 - 27*x^8 + 56*x^7 + 222*x^6 - 622*x^5 - 1890*x^4 + 2792*x^3 + 11345*x^2 - 225*x + 2061);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - x^9 - 27*x^8 + 56*x^7 + 222*x^6 - 622*x^5 - 1890*x^4 + 2792*x^3 + 11345*x^2 - 225*x + 2061)
 

\( x^{10} - x^{9} - 27x^{8} + 56x^{7} + 222x^{6} - 622x^{5} - 1890x^{4} + 2792x^{3} + 11345x^{2} - 225x + 2061 \) Copy content Toggle raw display

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:  $[2, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(2993701399765540096\) \(\medspace = 2^{8}\cdot 61^{9}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(70.41\)
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:  $2\cdot 61^{9/10}\approx 80.87733776568385$
Ramified primes:   \(2\), \(61\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{61}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{6}+\frac{1}{8}a^{5}+\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{3}{8}a^{2}+\frac{1}{8}a+\frac{1}{8}$, $\frac{1}{24}a^{7}+\frac{1}{6}a^{5}-\frac{1}{8}a^{4}-\frac{5}{24}a^{3}+\frac{1}{6}a-\frac{3}{8}$, $\frac{1}{48}a^{8}-\frac{1}{48}a^{7}-\frac{1}{24}a^{6}-\frac{1}{48}a^{5}-\frac{1}{6}a^{4}+\frac{5}{48}a^{3}-\frac{7}{24}a^{2}-\frac{19}{48}a+\frac{5}{16}$, $\frac{1}{198802947504}a^{9}-\frac{1317416777}{198802947504}a^{8}+\frac{23215763}{3550052634}a^{7}+\frac{343854955}{28400421072}a^{6}-\frac{14682008797}{99401473752}a^{5}+\frac{40687180165}{198802947504}a^{4}+\frac{23206823785}{49700736876}a^{3}+\frac{94256460259}{198802947504}a^{2}+\frac{894765777}{9466807024}a-\frac{6497359375}{16566912292}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$

Class group and class number

$C_{5}$, which has order $5$ (assuming GRH)

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:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{281422009}{49700736876}a^{9}-\frac{163189399}{66267649168}a^{8}-\frac{1488960029}{9466807024}a^{7}+\frac{136366986}{591675439}a^{6}+\frac{97664609513}{66267649168}a^{5}-\frac{47292947839}{16566912292}a^{4}-\frac{870112702717}{66267649168}a^{3}+\frac{177815526225}{16566912292}a^{2}+\frac{2163932778959}{28400421072}a+\frac{1969918989711}{66267649168}$, $\frac{7723165}{99401473752}a^{9}+\frac{25785617}{99401473752}a^{8}-\frac{95689589}{14200210536}a^{7}+\frac{40233826}{1775026317}a^{6}+\frac{4779479995}{99401473752}a^{5}-\frac{55696944565}{99401473752}a^{4}+\frac{148097811013}{99401473752}a^{3}-\frac{59705340293}{49700736876}a^{2}-\frac{224307429}{2366701756}a-\frac{2824567009}{16566912292}$, $\frac{260}{530243}a^{9}-\frac{1875}{2120972}a^{8}-\frac{6605}{454494}a^{7}+\frac{17925}{302996}a^{6}+\frac{151775}{1590729}a^{5}-\frac{957015}{1060486}a^{4}+\frac{210715}{3181458}a^{3}+\frac{8594675}{2120972}a^{2}-\frac{57415}{227247}a-\frac{81219849}{2120972}$, $\frac{154283125}{49700736876}a^{9}+\frac{2165394521}{198802947504}a^{8}-\frac{921792585}{9466807024}a^{7}-\frac{1577343421}{7100105268}a^{6}+\frac{92660762487}{66267649168}a^{5}+\frac{154503804019}{99401473752}a^{4}-\frac{868267455689}{66267649168}a^{3}-\frac{458335671695}{24850368438}a^{2}+\frac{1865266048673}{28400421072}a+\frac{9818848261743}{66267649168}$, $\frac{59117805}{33133824584}a^{9}-\frac{257047869}{66267649168}a^{8}-\frac{1439266973}{28400421072}a^{7}+\frac{890531643}{4733403512}a^{6}+\frac{57605231155}{198802947504}a^{5}-\frac{19815355673}{8283456146}a^{4}+\frac{113620292371}{198802947504}a^{3}+\frac{318136362321}{33133824584}a^{2}-\frac{59806242695}{28400421072}a+\frac{190180844193}{66267649168}$ Copy content Toggle raw display (assuming GRH)
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:  \( 176602.4324539309 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 176602.4324539309 \cdot 5}{2\cdot\sqrt{2993701399765540096}}\cr\approx \mathstrut & 1.59078656435450 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 - 27*x^8 + 56*x^7 + 222*x^6 - 622*x^5 - 1890*x^4 + 2792*x^3 + 11345*x^2 - 225*x + 2061)
 
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 - 27*x^8 + 56*x^7 + 222*x^6 - 622*x^5 - 1890*x^4 + 2792*x^3 + 11345*x^2 - 225*x + 2061, 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 - 27*x^8 + 56*x^7 + 222*x^6 - 622*x^5 - 1890*x^4 + 2792*x^3 + 11345*x^2 - 225*x + 2061);
 
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 - 27*x^8 + 56*x^7 + 222*x^6 - 622*x^5 - 1890*x^4 + 2792*x^3 + 11345*x^2 - 225*x + 2061);
 
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

$D_{10}$ (as 10T3):

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 solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{61}) \), 5.1.221533456.1

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

Galois closure: 20.0.143395969135330465829722243945339027456.2
Degree 10 sibling: 10.0.11974805599062160384.2
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 R ${\href{/padicField/3.2.0.1}{2} }^{4}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.5.0.1}{5} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{5}$ ${\href{/padicField/11.2.0.1}{2} }^{5}$ ${\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{5}$ ${\href{/padicField/29.10.0.1}{10} }$ ${\href{/padicField/31.2.0.1}{2} }^{5}$ ${\href{/padicField/37.2.0.1}{2} }^{5}$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{5}$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.2.0.1}{2} }^{5}$

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]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
\(61\) Copy content Toggle raw display 61.10.9.3$x^{10} + 976$$10$$1$$9$$C_{10}$$[\ ]_{10}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.61.2t1.a.a$1$ $ 61 $ \(\Q(\sqrt{61}) \) $C_2$ (as 2T1) $1$ $1$
1.244.2t1.a.a$1$ $ 2^{2} \cdot 61 $ \(\Q(\sqrt{-61}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.14884.10t3.a.b$2$ $ 2^{2} \cdot 61^{2}$ 10.2.2993701399765540096.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.14884.5t2.a.b$2$ $ 2^{2} \cdot 61^{2}$ 5.1.221533456.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.14884.5t2.a.a$2$ $ 2^{2} \cdot 61^{2}$ 5.1.221533456.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.14884.10t3.a.a$2$ $ 2^{2} \cdot 61^{2}$ 10.2.2993701399765540096.1 $D_{10}$ (as 10T3) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.