Properties

Label 8.4.85350443984.2
Degree 88
Signature [4,2][4, 2]
Discriminant 8535044398485350443984
Root discriminant 23.2523.25
Ramified primes 2,13,172,13,17
Class number 22
Class group [2]
Galois group ((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 (as 8T27)

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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 - 11*x^6 + 14*x^5 + 19*x^4 - 14*x^3 - 45*x^2 + 2*x + 52)
 
gp: K = bnfinit(y^8 - 2*y^7 - 11*y^6 + 14*y^5 + 19*y^4 - 14*y^3 - 45*y^2 + 2*y + 52, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^7 - 11*x^6 + 14*x^5 + 19*x^4 - 14*x^3 - 45*x^2 + 2*x + 52);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 2*x^7 - 11*x^6 + 14*x^5 + 19*x^4 - 14*x^3 - 45*x^2 + 2*x + 52)
 

x82x711x6+14x5+19x414x345x2+2x+52 x^{8} - 2x^{7} - 11x^{6} + 14x^{5} + 19x^{4} - 14x^{3} - 45x^{2} + 2x + 52 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:  [4,2][4, 2]
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   8535044398485350443984 =2413177\medspace = 2^{4}\cdot 13\cdot 17^{7} 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:  23.2523.25
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:  2131/2177/886.028921423413992\cdot 13^{1/2}17^{7/8}\approx 86.02892142341399
Ramified primes:   22, 1313, 1717 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(221)\Q(\sqrt{221})
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}, 12a312a\frac{1}{2}a^{3}-\frac{1}{2}a, 12a412a2\frac{1}{2}a^{4}-\frac{1}{2}a^{2}, 14a514a414a314a212a\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a, 18a614a414a3+18a214a12\frac{1}{8}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}, 1824a7+25412a6+7412a541412a4+81824a3+71206a253206a13103\frac{1}{824}a^{7}+\frac{25}{412}a^{6}+\frac{7}{412}a^{5}-\frac{41}{412}a^{4}+\frac{81}{824}a^{3}+\frac{71}{206}a^{2}-\frac{53}{206}a-\frac{13}{103} 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:  22

Class group and class number

C2C_{2}, which has order 22

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:  55
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:   13824a771824a63103a5+291412a4595824a3119824a239412a+177206\frac{13}{824}a^{7}-\frac{71}{824}a^{6}-\frac{3}{103}a^{5}+\frac{291}{412}a^{4}-\frac{595}{824}a^{3}-\frac{119}{824}a^{2}-\frac{39}{412}a+\frac{177}{206}, 21412a763824a6221412a5+33103a4+259412a3+83824a2847412a165206\frac{21}{412}a^{7}-\frac{63}{824}a^{6}-\frac{221}{412}a^{5}+\frac{33}{103}a^{4}+\frac{259}{412}a^{3}+\frac{83}{824}a^{2}-\frac{847}{412}a-\frac{165}{206}, 3824a77103a6+21412a5+289412a4169824a3295412a2131103a+64103\frac{3}{824}a^{7}-\frac{7}{103}a^{6}+\frac{21}{412}a^{5}+\frac{289}{412}a^{4}-\frac{169}{824}a^{3}-\frac{295}{412}a^{2}-\frac{131}{103}a+\frac{64}{103}, 13824a771824a63103a5+291412a4595824a3119824a2451412a+383206\frac{13}{824}a^{7}-\frac{71}{824}a^{6}-\frac{3}{103}a^{5}+\frac{291}{412}a^{4}-\frac{595}{824}a^{3}-\frac{119}{824}a^{2}-\frac{451}{412}a+\frac{383}{206}, 3412a79824a661412a5+83206a4+37412a3871824a2+85412a+153206\frac{3}{412}a^{7}-\frac{9}{824}a^{6}-\frac{61}{412}a^{5}+\frac{83}{206}a^{4}+\frac{37}{412}a^{3}-\frac{871}{824}a^{2}+\frac{85}{412}a+\frac{153}{206} 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:  230.308522105 230.308522105
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

lims1(s1)ζK(s)=(2r1(2π)r2RhwD(24(2π)2230.3085221052285350443984(0.497951259876 \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^{4}\cdot(2\pi)^{2}\cdot 230.308522105 \cdot 2}{2\cdot\sqrt{85350443984}}\cr\approx \mathstrut & 0.497951259876 \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 - 11*x^6 + 14*x^5 + 19*x^4 - 14*x^3 - 45*x^2 + 2*x + 52)
 
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 - 11*x^6 + 14*x^5 + 19*x^4 - 14*x^3 - 45*x^2 + 2*x + 52, 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 - 11*x^6 + 14*x^5 + 19*x^4 - 14*x^3 - 45*x^2 + 2*x + 52);
 
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 - 11*x^6 + 14*x^5 + 19*x^4 - 14*x^3 - 45*x^2 + 2*x + 52);
 
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

C2C4C_2\wr C_4 (as 8T27):

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 64
The 13 conjugacy class representatives for ((C8:C2):C2):C2((C_8 : C_2):C_2):C_2
Character table for ((C8:C2):C2):C2((C_8 : C_2):C_2):C_2

Intermediate fields

Q(17)\Q(\sqrt{17}) , 4.4.4913.1

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 32 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

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} } 42{\href{/padicField/5.4.0.1}{4} }^{2} 42{\href{/padicField/7.4.0.1}{4} }^{2} 42{\href{/padicField/11.4.0.1}{4} }^{2} R R 4,22{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2} 8{\href{/padicField/23.8.0.1}{8} } 8{\href{/padicField/29.8.0.1}{8} } 42{\href{/padicField/31.4.0.1}{4} }^{2} 42{\href{/padicField/37.4.0.1}{4} }^{2} 42{\href{/padicField/41.4.0.1}{4} }^{2} 24{\href{/padicField/43.2.0.1}{2} }^{4} 23,12{\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2} 24{\href{/padicField/53.2.0.1}{2} }^{4} 4,22{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{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 [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 2.2.1.0a1.1x2+x+1x^{2} + x + 1112200C2C_2[ ]2[\ ]^{2}
2.2.1.0a1.1x2+x+1x^{2} + x + 1112200C2C_2[ ]2[\ ]^{2}
2.2.2.4a1.2x4+2x3+5x2+8x+5x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5222244C4C_4[2]2[2]^{2}
1313 Copy content Toggle raw display Q13\Q_{13}x+11x + 11111100Trivial[ ][\ ]
Q13\Q_{13}x+11x + 11111100Trivial[ ][\ ]
13.1.2.1a1.1x2+13x^{2} + 13221111C2C_2[ ]2[\ ]_{2}
13.2.1.0a1.1x2+12x+2x^{2} + 12 x + 2112200C2C_2[ ]2[\ ]^{2}
13.2.1.0a1.1x2+12x+2x^{2} + 12 x + 2112200C2C_2[ ]2[\ ]^{2}
1717 Copy content Toggle raw display 17.1.8.7a1.7x8+221x^{8} + 221881177C8C_8[ ]8[\ ]_{8}

Spectrum of ring of integers

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