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Properties

Label 12.8.123...625.1
Degree $12$
Signature $[8, 2]$
Discriminant $1.230\times 10^{18}$
Root discriminant \(32.17\)
Ramified primes $5,229$
Class number $2$
Class group [2]
Galois group $C_4:S_4$ (as 12T54)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 14*x^10 + 37*x^9 - 20*x^8 - 276*x^7 + 87*x^6 + 408*x^5 + 69*x^4 + 72*x^3 - 143*x^2 - 120*x + 16)
 
Copy content gp:K = bnfinit(y^12 - y^11 - 14*y^10 + 37*y^9 - 20*y^8 - 276*y^7 + 87*y^6 + 408*y^5 + 69*y^4 + 72*y^3 - 143*y^2 - 120*y + 16, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 14*x^10 + 37*x^9 - 20*x^8 - 276*x^7 + 87*x^6 + 408*x^5 + 69*x^4 + 72*x^3 - 143*x^2 - 120*x + 16);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 14*x^10 + 37*x^9 - 20*x^8 - 276*x^7 + 87*x^6 + 408*x^5 + 69*x^4 + 72*x^3 - 143*x^2 - 120*x + 16)
 

\( x^{12} - x^{11} - 14 x^{10} + 37 x^{9} - 20 x^{8} - 276 x^{7} + 87 x^{6} + 408 x^{5} + 69 x^{4} + \cdots + 16 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $12$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[8, 2]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(1230006625291015625\) \(\medspace = 5^{9}\cdot 229^{5}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(32.17\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{3/4}229^{1/2}\approx 50.59938571007813$
Ramified primes:   \(5\), \(229\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{1145}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{609068293233088}a^{11}+\frac{45451745559707}{609068293233088}a^{10}-\frac{58045938568877}{304534146616544}a^{9}-\frac{269686829958963}{609068293233088}a^{8}-\frac{2257897836173}{76133536654136}a^{7}+\frac{10863157584803}{152267073308272}a^{6}-\frac{146358307711833}{609068293233088}a^{5}-\frac{19303159184265}{152267073308272}a^{4}+\frac{14481806263445}{609068293233088}a^{3}+\frac{27839717261093}{152267073308272}a^{2}+\frac{196468786339937}{609068293233088}a+\frac{26868152641673}{152267073308272}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

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

Class group and class number

Ideal class group:  $C_{2}$, which has order $2$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}\times C_{2}$, which has order $4$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $9$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{655956019}{87146700992}a^{11}-\frac{175337983}{87146700992}a^{10}-\frac{4877350903}{43573350496}a^{9}+\frac{17615889495}{87146700992}a^{8}+\frac{757974301}{10893337624}a^{7}-\frac{48508468863}{21786675248}a^{6}-\frac{76248945627}{87146700992}a^{5}+\frac{85416239253}{21786675248}a^{4}+\frac{235841650607}{87146700992}a^{3}+\frac{1341294215}{21786675248}a^{2}-\frac{10021767117}{87146700992}a-\frac{13300753797}{21786675248}$, $\frac{11345184520}{732053237059}a^{11}-\frac{10336433390}{732053237059}a^{10}-\frac{156853474395}{732053237059}a^{9}+\frac{411253270915}{732053237059}a^{8}-\frac{247792006435}{732053237059}a^{7}-\frac{3136072922750}{732053237059}a^{6}+\frac{1038670274126}{732053237059}a^{5}+\frac{3248382466375}{732053237059}a^{4}+\frac{101600917120}{732053237059}a^{3}+\frac{3576850133870}{732053237059}a^{2}-\frac{1134933236880}{732053237059}a+\frac{476695039089}{732053237059}$, $\frac{1136574055}{87146700992}a^{11}-\frac{746655523}{87146700992}a^{10}-\frac{8204592507}{43573350496}a^{9}+\frac{36798804283}{87146700992}a^{8}-\frac{865777475}{10893337624}a^{7}-\frac{81837748683}{21786675248}a^{6}-\frac{2214044367}{87146700992}a^{5}+\frac{133061410577}{21786675248}a^{4}+\frac{193977994099}{87146700992}a^{3}+\frac{22286280115}{21786675248}a^{2}-\frac{71656761017}{87146700992}a-\frac{59497928369}{21786675248}$, $\frac{575205656191}{304534146616544}a^{11}+\frac{894168076277}{304534146616544}a^{10}-\frac{3976997087235}{152267073308272}a^{9}+\frac{863391567379}{304534146616544}a^{8}+\frac{2530674561287}{38066768327068}a^{7}-\frac{37841326850183}{76133536654136}a^{6}-\frac{345711406820727}{304534146616544}a^{5}-\frac{28170213148183}{76133536654136}a^{4}+\frac{460952993380331}{304534146616544}a^{3}+\frac{120544292370719}{76133536654136}a^{2}+\frac{272437313171119}{304534146616544}a-\frac{675607756449}{76133536654136}$, $\frac{1568145489037}{609068293233088}a^{11}-\frac{4476109599681}{609068293233088}a^{10}-\frac{12000396970825}{304534146616544}a^{9}+\frac{100275169238377}{609068293233088}a^{8}-\frac{9595158620149}{76133536654136}a^{7}-\frac{126118908766161}{152267073308272}a^{6}+\frac{10\cdots 71}{609068293233088}a^{5}+\frac{381362329107563}{152267073308272}a^{4}-\frac{221674135796207}{609068293233088}a^{3}-\frac{32159197396343}{152267073308272}a^{2}-\frac{10\cdots 15}{609068293233088}a-\frac{92622580960171}{152267073308272}$, $\frac{11220739766715}{609068293233088}a^{11}-\frac{24312653074215}{609068293233088}a^{10}-\frac{65192140008543}{304534146616544}a^{9}+\frac{563569890518143}{609068293233088}a^{8}-\frac{105555574731155}{76133536654136}a^{7}-\frac{530338777420343}{152267073308272}a^{6}+\frac{32\cdots 57}{609068293233088}a^{5}+\frac{469171929919709}{152267073308272}a^{4}-\frac{10\cdots 73}{609068293233088}a^{3}+\frac{46875767558639}{152267073308272}a^{2}-\frac{20\cdots 57}{609068293233088}a+\frac{54792191849619}{152267073308272}$, $\frac{6636909890525}{609068293233088}a^{11}-\frac{9548364513905}{609068293233088}a^{10}-\frac{41748828491001}{304534146616544}a^{9}+\frac{274922087864185}{609068293233088}a^{8}-\frac{39434946305793}{76133536654136}a^{7}-\frac{371117010938121}{152267073308272}a^{6}+\frac{940394796971819}{609068293233088}a^{5}+\frac{315538232072283}{152267073308272}a^{4}+\frac{390557558555169}{609068293233088}a^{3}+\frac{75392262414753}{152267073308272}a^{2}-\frac{604590147006915}{609068293233088}a-\frac{75238879018187}{152267073308272}$, $\frac{636700505629}{152267073308272}a^{11}-\frac{1418164522505}{152267073308272}a^{10}-\frac{3046518089817}{76133536654136}a^{9}+\frac{30657862023705}{152267073308272}a^{8}-\frac{4073969499628}{9516692081767}a^{7}-\frac{16768564289687}{38066768327068}a^{6}+\frac{128239857971795}{152267073308272}a^{5}-\frac{42014719644205}{38066768327068}a^{4}+\frac{114490719077793}{152267073308272}a^{3}+\frac{37799077767667}{38066768327068}a^{2}+\frac{138935078774053}{152267073308272}a+\frac{19699283275169}{38066768327068}$, $\frac{8033300600659}{609068293233088}a^{11}-\frac{17009358756479}{609068293233088}a^{10}-\frac{45980906675799}{304534146616544}a^{9}+\frac{395826752824247}{609068293233088}a^{8}-\frac{77088051425391}{76133536654136}a^{7}-\frac{362508614396503}{152267073308272}a^{6}+\frac{21\cdots 53}{609068293233088}a^{5}+\frac{203819779791829}{152267073308272}a^{4}-\frac{73385551894321}{609068293233088}a^{3}+\frac{137767062052255}{152267073308272}a^{2}-\frac{593724458252941}{609068293233088}a-\frac{147357603712885}{152267073308272}$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 37171.8222915 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{2}\cdot 37171.8222915 \cdot 2}{2\cdot\sqrt{1230006625291015625}}\cr\approx \mathstrut & 0.338734809844 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 14*x^10 + 37*x^9 - 20*x^8 - 276*x^7 + 87*x^6 + 408*x^5 + 69*x^4 + 72*x^3 - 143*x^2 - 120*x + 16) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^12 - x^11 - 14*x^10 + 37*x^9 - 20*x^8 - 276*x^7 + 87*x^6 + 408*x^5 + 69*x^4 + 72*x^3 - 143*x^2 - 120*x + 16, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 14*x^10 + 37*x^9 - 20*x^8 - 276*x^7 + 87*x^6 + 408*x^5 + 69*x^4 + 72*x^3 - 143*x^2 - 120*x + 16); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 - 14*x^10 + 37*x^9 - 20*x^8 - 276*x^7 + 87*x^6 + 408*x^5 + 69*x^4 + 72*x^3 - 143*x^2 - 120*x + 16); 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

$C_4:S_4$ (as 12T54):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 96
The 14 conjugacy class representatives for $C_4:S_4$
Character table for $C_4:S_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.229.1, 6.6.6555125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 12 sibling: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
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 ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ ${\href{/padicField/3.12.0.1}{12} }$ R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
\(5\) Copy content Toggle raw display 5.3.4.9a1.1$x^{12} + 12 x^{10} + 12 x^{9} + 54 x^{8} + 108 x^{7} + 162 x^{6} + 324 x^{5} + 405 x^{4} + 432 x^{3} + 491 x^{2} + 324 x + 81$$4$$3$$9$$C_{12}$$$[\ ]_{4}^{3}$$
\(229\) Copy content Toggle raw display $\Q_{229}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{229}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$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)