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Properties

Label 10.0.8784925479251312.1
Degree $10$
Signature $[0, 5]$
Discriminant $-8.785\times 10^{15}$
Root discriminant \(39.30\)
Ramified primes $2,887$
Class number $29$
Class group [29]
Galois group $A_5\times C_2$ (as 10T11)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 18*x^8 - 18*x^7 + 85*x^6 - 174*x^5 + 117*x^4 - 144*x^3 + 112*x^2 - 24*x + 923)
 
gp: K = bnfinit(y^10 + 18*y^8 - 18*y^7 + 85*y^6 - 174*y^5 + 117*y^4 - 144*y^3 + 112*y^2 - 24*y + 923, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 + 18*x^8 - 18*x^7 + 85*x^6 - 174*x^5 + 117*x^4 - 144*x^3 + 112*x^2 - 24*x + 923);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 + 18*x^8 - 18*x^7 + 85*x^6 - 174*x^5 + 117*x^4 - 144*x^3 + 112*x^2 - 24*x + 923)
 

\( x^{10} + 18x^{8} - 18x^{7} + 85x^{6} - 174x^{5} + 117x^{4} - 144x^{3} + 112x^{2} - 24x + 923 \) 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:  $[0, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-8784925479251312\) \(\medspace = -\,2^{4}\cdot 887^{5}\) 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:  \(39.30\)
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:  $2^{2/3}887^{1/2}\approx 47.2768436183456$
Ramified primes:   \(2\), \(887\) 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:  \(\Q(\sqrt{-887}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{1}{8}$, $\frac{1}{1176801248}a^{9}-\frac{2003139}{1176801248}a^{8}+\frac{261998283}{1176801248}a^{7}+\frac{250761453}{1176801248}a^{6}-\frac{41350097}{588400624}a^{5}+\frac{61928323}{147100156}a^{4}-\frac{51596179}{1176801248}a^{3}+\frac{81714127}{168114464}a^{2}-\frac{22713805}{168114464}a+\frac{517076025}{1176801248}$ 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:  $2$

Class group and class number

$C_{29}$, which has order $29$

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:   \( -1 \)  (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:   $\frac{1314261}{1176801248}a^{9}+\frac{4026653}{1176801248}a^{8}+\frac{20846879}{1176801248}a^{7}+\frac{13575869}{1176801248}a^{6}+\frac{8899479}{588400624}a^{5}-\frac{21749951}{147100156}a^{4}-\frac{27495215}{1176801248}a^{3}+\frac{90245455}{168114464}a^{2}+\frac{175270759}{168114464}a-\frac{594096431}{1176801248}$, $\frac{123061}{84057232}a^{9}+\frac{51515}{84057232}a^{8}+\frac{1268947}{84057232}a^{7}+\frac{748795}{84057232}a^{6}-\frac{1126179}{42028616}a^{5}+\frac{864116}{5253577}a^{4}-\frac{46250335}{84057232}a^{3}+\frac{8986905}{12008176}a^{2}-\frac{13411189}{12008176}a+\frac{137295211}{84057232}$, $\frac{8541783}{1176801248}a^{9}+\frac{17288771}{1176801248}a^{8}+\frac{132027005}{1176801248}a^{7}+\frac{126148803}{1176801248}a^{6}+\frac{99970209}{588400624}a^{5}-\frac{42562643}{147100156}a^{4}-\frac{1882861173}{1176801248}a^{3}-\frac{57373935}{168114464}a^{2}+\frac{474740949}{168114464}a+\frac{3997883623}{1176801248}$, $\frac{965521}{147100156}a^{9}+\frac{80669}{147100156}a^{8}+\frac{15680753}{147100156}a^{7}-\frac{7651897}{147100156}a^{6}+\frac{14779692}{36775039}a^{5}-\frac{19143843}{36775039}a^{4}-\frac{34863065}{147100156}a^{3}-\frac{32262119}{21014308}a^{2}-\frac{29324219}{21014308}a-\frac{460119679}{147100156}$ Copy content Toggle raw display
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:  \( 2373.5998999867606 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 2373.5998999867606 \cdot 29}{2\cdot\sqrt{8784925479251312}}\cr\approx \mathstrut & 3.59588519885281 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 + 18*x^8 - 18*x^7 + 85*x^6 - 174*x^5 + 117*x^4 - 144*x^3 + 112*x^2 - 24*x + 923)
 
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 + 18*x^8 - 18*x^7 + 85*x^6 - 174*x^5 + 117*x^4 - 144*x^3 + 112*x^2 - 24*x + 923, 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 + 18*x^8 - 18*x^7 + 85*x^6 - 174*x^5 + 117*x^4 - 144*x^3 + 112*x^2 - 24*x + 923);
 
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 + 18*x^8 - 18*x^7 + 85*x^6 - 174*x^5 + 117*x^4 - 144*x^3 + 112*x^2 - 24*x + 923);
 
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

$C_2\times A_5$ (as 10T11):

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 120
The 10 conjugacy class representatives for $A_5\times C_2$
Character table for $A_5\times C_2$

Intermediate fields

\(\Q(\sqrt{-887}) \), 5.1.3147076.1

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

Degree 12 siblings: 12.4.158465397596416.1, deg 12
Degree 20 siblings: deg 20, deg 20
Degree 24 sibling: deg 24
Degree 30 siblings: deg 30, deg 30
Degree 40 sibling: deg 40
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 ${\href{/padicField/3.5.0.1}{5} }^{2}$ ${\href{/padicField/5.10.0.1}{10} }$ ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }$ ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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$$[\ ]^{2}$
2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
\(887\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.887.2t1.a.a$1$ $ 887 $ \(\Q(\sqrt{-887}) \) $C_2$ (as 2T1) $1$ $-1$
3.3548.12t76.a.a$3$ $ 2^{2} \cdot 887 $ 10.0.8784925479251312.1 $A_5\times C_2$ (as 10T11) $1$ $1$
3.3147076.12t33.a.a$3$ $ 2^{2} \cdot 887^{2}$ 5.1.3147076.1 $A_5$ (as 5T4) $1$ $-1$
3.3147076.12t33.a.b$3$ $ 2^{2} \cdot 887^{2}$ 5.1.3147076.1 $A_5$ (as 5T4) $1$ $-1$
3.3548.12t76.a.b$3$ $ 2^{2} \cdot 887 $ 10.0.8784925479251312.1 $A_5\times C_2$ (as 10T11) $1$ $1$
* 4.3147076.5t4.a.a$4$ $ 2^{2} \cdot 887^{2}$ 5.1.3147076.1 $A_5$ (as 5T4) $1$ $0$
* 4.3147076.10t11.a.a$4$ $ 2^{2} \cdot 887^{2}$ 10.0.8784925479251312.1 $A_5\times C_2$ (as 10T11) $1$ $0$
5.12588304.6t12.a.a$5$ $ 2^{4} \cdot 887^{2}$ 5.1.3147076.1 $A_5$ (as 5T4) $1$ $1$
5.11165825648.12t75.a.a$5$ $ 2^{4} \cdot 887^{3}$ 10.0.8784925479251312.1 $A_5\times C_2$ (as 10T11) $1$ $-1$

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 *.