Properties

Label 8.6.307235186944.2
Degree 88
Signature [6,1][6, 1]
Discriminant 307235186944-307235186944
Root discriminant 27.2927.29
Ramified primes 2,7,1012,7,101
Class number 11
Class group trivial
Galois group C2S4C_2 \wr S_4 (as 8T44)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 3*x^6 - 6*x^5 + 8*x^4 + 26*x^3 - 26*x^2 + 2*x + 2)
 
gp: K = bnfinit(y^8 - 2*y^7 - 3*y^6 - 6*y^5 + 8*y^4 + 26*y^3 - 26*y^2 + 2*y + 2, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^7 - 3*x^6 - 6*x^5 + 8*x^4 + 26*x^3 - 26*x^2 + 2*x + 2);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 2*x^7 - 3*x^6 - 6*x^5 + 8*x^4 + 26*x^3 - 26*x^2 + 2*x + 2)
 

x82x73x66x5+8x4+26x326x2+2x+2 x^{8} - 2x^{7} - 3x^{6} - 6x^{5} + 8x^{4} + 26x^{3} - 26x^{2} + 2x + 2 Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  88
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  [6,1][6, 1]
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   307235186944-307235186944 =28761012\medspace = -\,2^{8}\cdot 7^{6}\cdot 101^{2} 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:  27.2927.29
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:  219/1273/41011/2129.602996669240072^{19/12}7^{3/4}101^{1/2}\approx 129.60299666924007
Ramified primes:   22, 77, 101101 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(1)\Q(\sqrt{-1})
Aut(K/Q)\Aut(K/\Q):   C2C_2
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over Q\Q.
This is not a CM field.

Integral basis (with respect to field generator aa)

11, aa, a2a^{2}, a3a^{3}, a4a^{4}, a5a^{5}, a6a^{6}, 1127a748127a6+46127a5+37127a443127a328127a28127a11127\frac{1}{127}a^{7}-\frac{48}{127}a^{6}+\frac{46}{127}a^{5}+\frac{37}{127}a^{4}-\frac{43}{127}a^{3}-\frac{28}{127}a^{2}-\frac{8}{127}a-\frac{11}{127} Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  11
Inessential primes:  None

Class group and class number

Trivial group, which has order 11

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:  66
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   1 -1  (order 22) 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:   46127a749127a6170127a5457127a473127a3+998127a2241127a125127\frac{46}{127}a^{7}-\frac{49}{127}a^{6}-\frac{170}{127}a^{5}-\frac{457}{127}a^{4}-\frac{73}{127}a^{3}+\frac{998}{127}a^{2}-\frac{241}{127}a-\frac{125}{127}, 199127a7281127a6752127a51654127a4+587127a3+5477127a21846127a411127\frac{199}{127}a^{7}-\frac{281}{127}a^{6}-\frac{752}{127}a^{5}-\frac{1654}{127}a^{4}+\frac{587}{127}a^{3}+\frac{5477}{127}a^{2}-\frac{1846}{127}a-\frac{411}{127}, 117127a7155127a6460127a51005127a4+303127a3+3201127a21063127a271127\frac{117}{127}a^{7}-\frac{155}{127}a^{6}-\frac{460}{127}a^{5}-\frac{1005}{127}a^{4}+\frac{303}{127}a^{3}+\frac{3201}{127}a^{2}-\frac{1063}{127}a-\frac{271}{127}, 82127a7126127a6292127a5649127a4+284127a3+2276127a21037127a267127\frac{82}{127}a^{7}-\frac{126}{127}a^{6}-\frac{292}{127}a^{5}-\frac{649}{127}a^{4}+\frac{284}{127}a^{3}+\frac{2276}{127}a^{2}-\frac{1037}{127}a-\frac{267}{127}, 47127a797127a6124127a5293127a4+392127a3+1097127a21138127a+245127\frac{47}{127}a^{7}-\frac{97}{127}a^{6}-\frac{124}{127}a^{5}-\frac{293}{127}a^{4}+\frac{392}{127}a^{3}+\frac{1097}{127}a^{2}-\frac{1138}{127}a+\frac{245}{127}, 199127a7281127a6752127a51654127a4+587127a3+5477127a21846127a665127\frac{199}{127}a^{7}-\frac{281}{127}a^{6}-\frac{752}{127}a^{5}-\frac{1654}{127}a^{4}+\frac{587}{127}a^{3}+\frac{5477}{127}a^{2}-\frac{1846}{127}a-\frac{665}{127} 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:  2009.20325773 2009.20325773
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

lims1(s1)ζK(s)=(2r1(2π)r2RhwD(26(2π)12009.2032577312307235186944(0.728816579864 \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^{6}\cdot(2\pi)^{1}\cdot 2009.20325773 \cdot 1}{2\cdot\sqrt{307235186944}}\cr\approx \mathstrut & 0.728816579864 \end{aligned}

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 3*x^6 - 6*x^5 + 8*x^4 + 26*x^3 - 26*x^2 + 2*x + 2)
 
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^8 - 2*x^7 - 3*x^6 - 6*x^5 + 8*x^4 + 26*x^3 - 26*x^2 + 2*x + 2, 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(Rationals()); K<a> := NumberField(x^8 - 2*x^7 - 3*x^6 - 6*x^5 + 8*x^4 + 26*x^3 - 26*x^2 + 2*x + 2);
 
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^8 - 2*x^7 - 3*x^6 - 6*x^5 + 8*x^4 + 26*x^3 - 26*x^2 + 2*x + 2);
 
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

C2S4C_2\wr S_4 (as 8T44):

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 384
The 20 conjugacy class representatives for C2S4C_2 \wr S_4
Character table for C2S4C_2 \wr S_4

Intermediate fields

4.4.19796.1

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 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 8.2.307235186944.1

Frobenius cycle types

pp 22 33 55 77 1111 1313 1717 1919 2323 2929 3131 3737 4141 4343 4747 5353 5959
Cycle type R 8{\href{/padicField/3.8.0.1}{8} } 6,2{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} } R 8{\href{/padicField/11.8.0.1}{8} } 6,2{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} } 6,2{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} } 6,12{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2} 32,2{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} } 42{\href{/padicField/29.4.0.1}{4} }^{2} 32,2{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} } 24{\href{/padicField/37.2.0.1}{2} }^{4} 42{\href{/padicField/41.4.0.1}{4} }^{2} 4,22{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2} 32,2{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} } 24{\href{/padicField/53.2.0.1}{2} }^{4} 8{\href{/padicField/59.8.0.1}{8} }

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 [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\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 [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=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

ppLabelPolynomial ee ff cc Galois group Slope content
22 Copy content Toggle raw display 2.1.2.2a1.2x2+2x+6x^{2} + 2 x + 6221122C2C_2[2][2]
2.1.6.6a1.1x6+2x+2x^{6} + 2 x + 2661166S4S_4[43,43]32[\frac{4}{3}, \frac{4}{3}]_{3}^{2}
77 Copy content Toggle raw display 7.2.4.6a1.1x8+24x7+228x6+1080x5+2646x4+3240x3+2052x2+655x+81x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2646 x^{4} + 3240 x^{3} + 2052 x^{2} + 655 x + 81442266C8:C2C_8:C_2[ ]44[\ ]_{4}^{4}
101101 Copy content Toggle raw display 101.2.1.0a1.1x2+97x+2x^{2} + 97 x + 2112200C2C_2[ ]2[\ ]^{2}
101.2.1.0a1.1x2+97x+2x^{2} + 97 x + 2112200C2C_2[ ]2[\ ]^{2}
101.2.2.2a1.2x4+194x3+9413x2+388x+105x^{4} + 194 x^{3} + 9413 x^{2} + 388 x + 105222222C22C_2^2[ ]22[\ ]_{2}^{2}

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)