Properties

Label 12.2.449...576.9
Degree 1212
Signature [2,5][2, 5]
Discriminant 4.492×1018-4.492\times 10^{18}
Root discriminant 35.8435.84
Ramified primes 2,172,17
Class number 22
Class group [2]
Galois group S42:D4S_4^2:D_4 (as 12T260)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 34*x^9 - 34*x^8 - 34*x^7 + 187*x^6 + 170*x^5 - 119*x^4 - 986*x^3 - 1071*x^2 - 272*x + 544)
 
gp: K = bnfinit(y^12 - 34*y^9 - 34*y^8 - 34*y^7 + 187*y^6 + 170*y^5 - 119*y^4 - 986*y^3 - 1071*y^2 - 272*y + 544, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 34*x^9 - 34*x^8 - 34*x^7 + 187*x^6 + 170*x^5 - 119*x^4 - 986*x^3 - 1071*x^2 - 272*x + 544);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 34*x^9 - 34*x^8 - 34*x^7 + 187*x^6 + 170*x^5 - 119*x^4 - 986*x^3 - 1071*x^2 - 272*x + 544)
 

x1234x934x834x7+187x6+170x5119x4986x31071x2272x+544 x^{12} - 34x^{9} - 34x^{8} - 34x^{7} + 187x^{6} + 170x^{5} - 119x^{4} - 986x^{3} - 1071x^{2} - 272x + 544 Copy content Toggle raw display

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

Invariants

Degree:  1212
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  [2,5][2, 5]
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   4492085992834072576-4492085992834072576 =2171711\medspace = -\,2^{17}\cdot 17^{11} 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:  35.8435.84
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:  219/81711/1269.639837807334492^{19/8}17^{11/12}\approx 69.63983780733449
Ramified primes:   22, 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(34)\Q(\sqrt{-34})
#Aut(K/Q)\card{ \Aut(K/\Q) }:  22
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}, a7a^{7}, a8a^{8}, 12a912a712a512a\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a, 14a1014a9+14a8+14a7+14a6+14a512a4+14a2+14a\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}+\frac{1}{4}a, 1310084440827932a11+8461517356005310084440827932a10+21388349982331310084440827932a9+96241932080819310084440827932a8149974422985685310084440827932a7+111805106618203310084440827932a6+12891377893077521110206983a53394982870153177521110206983a438296180102719310084440827932a3130707787772401310084440827932a2+13305159320009155042220413966a1047259645105077521110206983\frac{1}{310084440827932}a^{11}+\frac{8461517356005}{310084440827932}a^{10}+\frac{21388349982331}{310084440827932}a^{9}+\frac{96241932080819}{310084440827932}a^{8}-\frac{149974422985685}{310084440827932}a^{7}+\frac{111805106618203}{310084440827932}a^{6}+\frac{128913778930}{77521110206983}a^{5}-\frac{33949828701531}{77521110206983}a^{4}-\frac{38296180102719}{310084440827932}a^{3}-\frac{130707787772401}{310084440827932}a^{2}+\frac{13305159320009}{155042220413966}a-\frac{10472596451050}{77521110206983} 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:  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:   1013589437039310084440827932a11407980810183155042220413966a101181972036279155042220413966a9727823209424677521110206983a8327986893150777521110206983a7+1774595920221477521110206983a6+147228156557051310084440827932a5+38606538085329155042220413966a4529809053356661310084440827932a319517820631852677521110206983a269749024987583310084440827932a+9732765848452377521110206983\frac{1013589437039}{310084440827932}a^{11}-\frac{407980810183}{155042220413966}a^{10}-\frac{1181972036279}{155042220413966}a^{9}-\frac{7278232094246}{77521110206983}a^{8}-\frac{3279868931507}{77521110206983}a^{7}+\frac{17745959202214}{77521110206983}a^{6}+\frac{147228156557051}{310084440827932}a^{5}+\frac{38606538085329}{155042220413966}a^{4}-\frac{529809053356661}{310084440827932}a^{3}-\frac{195178206318526}{77521110206983}a^{2}-\frac{69749024987583}{310084440827932}a+\frac{97327658484523}{77521110206983}, 977844997055155042220413966a1177213601854077521110206983a10+43184961248577521110206983a91587284781948977521110206983a8+834062959962877521110206983a7673104318441077521110206983a6+168366505496319155042220413966a56008154878734477521110206983a4106205820865321155042220413966a326362875798680777521110206983a246457132913899155042220413966a+9444731772569177521110206983\frac{977844997055}{155042220413966}a^{11}-\frac{772136018540}{77521110206983}a^{10}+\frac{431849612485}{77521110206983}a^{9}-\frac{15872847819489}{77521110206983}a^{8}+\frac{8340629599628}{77521110206983}a^{7}-\frac{6731043184410}{77521110206983}a^{6}+\frac{168366505496319}{155042220413966}a^{5}-\frac{60081548787344}{77521110206983}a^{4}-\frac{106205820865321}{155042220413966}a^{3}-\frac{263628757986807}{77521110206983}a^{2}-\frac{46457132913899}{155042220413966}a+\frac{94447317725691}{77521110206983}, 402248583643155042220413966a11+55328942439077521110206983a10166801625279577521110206983a9520329019481977521110206983a82479254182182077521110206983a7+2634198814292077521110206983a6+54803085923323155042220413966a5+9783320586316677521110206983a4295725128935517155042220413966a332075733746212777521110206983a2288038596773391155042220413966a+21881558498070377521110206983\frac{402248583643}{155042220413966}a^{11}+\frac{553289424390}{77521110206983}a^{10}-\frac{1668016252795}{77521110206983}a^{9}-\frac{5203290194819}{77521110206983}a^{8}-\frac{24792541821820}{77521110206983}a^{7}+\frac{26341988142920}{77521110206983}a^{6}+\frac{54803085923323}{155042220413966}a^{5}+\frac{97833205863166}{77521110206983}a^{4}-\frac{295725128935517}{155042220413966}a^{3}-\frac{320757337462127}{77521110206983}a^{2}-\frac{288038596773391}{155042220413966}a+\frac{218815584980703}{77521110206983}, 827855769499155042220413966a1184961658217977521110206983a10+123134971620677521110206983a91604086450522677521110206983a8+1765200357727577521110206983a72891159088164177521110206983a6+257752007291725155042220413966a514703221473760377521110206983a4+107376922363955155042220413966a342320720737822277521110206983a2+426091154715817155042220413966a+10039445638893777521110206983\frac{827855769499}{155042220413966}a^{11}-\frac{849616582179}{77521110206983}a^{10}+\frac{1231349716206}{77521110206983}a^{9}-\frac{16040864505226}{77521110206983}a^{8}+\frac{17652003577275}{77521110206983}a^{7}-\frac{28911590881641}{77521110206983}a^{6}+\frac{257752007291725}{155042220413966}a^{5}-\frac{147032214737603}{77521110206983}a^{4}+\frac{107376922363955}{155042220413966}a^{3}-\frac{423207207378222}{77521110206983}a^{2}+\frac{426091154715817}{155042220413966}a+\frac{100394456388937}{77521110206983}, 10408139013977521110206983a11+1638031874059310084440827932a10+733724584443310084440827932a917214501792165310084440827932a866718902670951310084440827932a780016342734493310084440827932a6+86712846239273310084440827932a5+150572993780539155042220413966a4138120029763277521110206983a310 ⁣ ⁣93310084440827932a215 ⁣ ⁣55310084440827932a22765569641499977521110206983\frac{104081390139}{77521110206983}a^{11}+\frac{1638031874059}{310084440827932}a^{10}+\frac{733724584443}{310084440827932}a^{9}-\frac{17214501792165}{310084440827932}a^{8}-\frac{66718902670951}{310084440827932}a^{7}-\frac{80016342734493}{310084440827932}a^{6}+\frac{86712846239273}{310084440827932}a^{5}+\frac{150572993780539}{155042220413966}a^{4}-\frac{1381200297632}{77521110206983}a^{3}-\frac{10\!\cdots\!93}{310084440827932}a^{2}-\frac{15\!\cdots\!55}{310084440827932}a-\frac{227655696414999}{77521110206983}, 831304609577521110206983a11489097509677521110206983a109375912989577521110206983a910657867818077521110206983a8+67400489868677521110206983a7+143882996403177521110206983a6704507622900377521110206983a53244635674757077521110206983a47113629353868077521110206983a36134280560257277521110206983a22387385274289277521110206983a+4003304423652977521110206983\frac{8313046095}{77521110206983}a^{11}-\frac{4890975096}{77521110206983}a^{10}-\frac{93759129895}{77521110206983}a^{9}-\frac{106578678180}{77521110206983}a^{8}+\frac{674004898686}{77521110206983}a^{7}+\frac{1438829964031}{77521110206983}a^{6}-\frac{7045076229003}{77521110206983}a^{5}-\frac{32446356747570}{77521110206983}a^{4}-\frac{71136293538680}{77521110206983}a^{3}-\frac{61342805602572}{77521110206983}a^{2}-\frac{23873852742892}{77521110206983}a+\frac{40033044236529}{77521110206983} 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:  112558.113937 112558.113937
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

lims1(s1)ζK(s)=(2r1(2π)r2RhwD(22(2π)5112558.113937224492085992834072576(2.08023361368 \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)^{5}\cdot 112558.113937 \cdot 2}{2\cdot\sqrt{4492085992834072576}}\cr\approx \mathstrut & 2.08023361368 \end{aligned}

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 34*x^9 - 34*x^8 - 34*x^7 + 187*x^6 + 170*x^5 - 119*x^4 - 986*x^3 - 1071*x^2 - 272*x + 544)
 
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^12 - 34*x^9 - 34*x^8 - 34*x^7 + 187*x^6 + 170*x^5 - 119*x^4 - 986*x^3 - 1071*x^2 - 272*x + 544, 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^12 - 34*x^9 - 34*x^8 - 34*x^7 + 187*x^6 + 170*x^5 - 119*x^4 - 986*x^3 - 1071*x^2 - 272*x + 544);
 
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^12 - 34*x^9 - 34*x^8 - 34*x^7 + 187*x^6 + 170*x^5 - 119*x^4 - 986*x^3 - 1071*x^2 - 272*x + 544);
 
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

S42:D4S_4^2:D_4 (as 12T260):

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 4608
The 65 conjugacy class representatives for S42:D4S_4^2:D_4
Character table for S42:D4S_4^2:D_4

Intermediate fields

Q(17)\Q(\sqrt{17}) , 6.2.11358856.1

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: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.2.561510749104259072.1

Frobenius cycle types

pp 22 33 55 77 1111 1313 1717 1919 2323 2929 3131 3737 4141 4343 4747 5353 5959
Cycle type R 43{\href{/padicField/3.4.0.1}{4} }^{3} 8,4{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} } 62{\href{/padicField/7.6.0.1}{6} }^{2} 8,22{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2} 4,22,14{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4} R 6,4,12{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2} 42,22{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2} 8,4{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} } 62{\href{/padicField/31.6.0.1}{6} }^{2} 8,4{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} } 12{\href{/padicField/41.12.0.1}{12} } 4,23,12{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2} 6,32{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2} 32,23{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3} 32,22,12{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{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 2.2.3.1x2+4x+2x^{2} + 4 x + 2221133C2C_2[3][3]
2.2.3.1x2+4x+2x^{2} + 4 x + 2221133C2C_2[3][3]
2.2.2.1x2+2x+2x^{2} + 2 x + 2221122C2C_2[2][2]
2.6.9.2x6+4x510x4+160x3+1212x2+2160x1048x^{6} + 4 x^{5} - 10 x^{4} + 160 x^{3} + 1212 x^{2} + 2160 x - 1048223399A4×C2A_4\times C_2[2,2,3]3[2, 2, 3]^{3}
1717 Copy content Toggle raw display 17.12.11.2x12+34x^{12} + 341212111111S3×C4S_3 \times C_4[ ]122[\ ]_{12}^{2}