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Properties

Label 12.4.311...801.1
Degree $12$
Signature $[4, 4]$
Discriminant $3.110\times 10^{20}$
Root discriminant \(51.02\)
Ramified primes $41,79$
Class number $1$
Class group trivial
Galois group $C_2\times A_5$ (as 12T76)

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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 - 4*x^11 + 2*x^10 + 13*x^9 - 55*x^8 + 5*x^7 - 21*x^6 - 8*x^5 - 285*x^4 + 557*x^3 - 490*x^2 + 224*x - 40)
 
Copy content gp:K = bnfinit(y^12 - 4*y^11 + 2*y^10 + 13*y^9 - 55*y^8 + 5*y^7 - 21*y^6 - 8*y^5 - 285*y^4 + 557*y^3 - 490*y^2 + 224*y - 40, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 2*x^10 + 13*x^9 - 55*x^8 + 5*x^7 - 21*x^6 - 8*x^5 - 285*x^4 + 557*x^3 - 490*x^2 + 224*x - 40);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 2*x^10 + 13*x^9 - 55*x^8 + 5*x^7 - 21*x^6 - 8*x^5 - 285*x^4 + 557*x^3 - 490*x^2 + 224*x - 40)
 

\( x^{12} - 4 x^{11} + 2 x^{10} + 13 x^{9} - 55 x^{8} + 5 x^{7} - 21 x^{6} - 8 x^{5} - 285 x^{4} + \cdots - 40 \) 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:  $[4, 4]$
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:   \(311013484453206508801\) \(\medspace = 41^{8}\cdot 79^{4}\) 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:  \(51.02\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $41^{4/5}79^{1/2}\approx 173.3966213773553$
Ramified primes:   \(41\), \(79\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{19551893516}a^{11}-\frac{1404110099}{9775946758}a^{10}+\frac{934082996}{4887973379}a^{9}-\frac{3363812447}{19551893516}a^{8}+\frac{3929541855}{19551893516}a^{7}-\frac{7520033081}{19551893516}a^{6}-\frac{4365281491}{19551893516}a^{5}+\frac{922742465}{9775946758}a^{4}+\frac{2992473711}{19551893516}a^{3}-\frac{1447986387}{19551893516}a^{2}-\frac{2015952581}{4887973379}a+\frac{79359731}{4887973379}$ 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:  Trivial group, which has order $1$
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:  Trivial group, which has order $1$
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:  $7$
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:   $\frac{194174991}{9775946758}a^{11}+\frac{351656354}{4887973379}a^{10}-\frac{75086241}{4887973379}a^{9}-\frac{1262594191}{4887973379}a^{8}+\frac{9875822203}{9775946758}a^{7}+\frac{2364598487}{9775946758}a^{6}+\frac{5738367969}{9775946758}a^{5}+\frac{3240144153}{4887973379}a^{4}+\frac{57111997377}{9775946758}a^{3}-\frac{83860294133}{9775946758}a^{2}+\frac{78741336923}{9775946758}a-\frac{13186400298}{4887973379}$, $\frac{3453034379}{19551893516}a^{11}+\frac{6066822963}{9775946758}a^{10}-\frac{136301636}{4887973379}a^{9}-\frac{46719175245}{19551893516}a^{8}+\frac{165951864421}{19551893516}a^{7}+\frac{73093303969}{19551893516}a^{6}+\frac{88927362803}{19551893516}a^{5}+\frac{10872361187}{4887973379}a^{4}+\frac{1028124568105}{19551893516}a^{3}-\frac{1383661223465}{19551893516}a^{2}+\frac{239262062980}{4887973379}a-\frac{67409939709}{4887973379}$, $\frac{11183844829}{19551893516}a^{11}-\frac{9834570747}{4887973379}a^{10}+\frac{1586886073}{9775946758}a^{9}+\frac{148020322543}{19551893516}a^{8}-\frac{544487282305}{19551893516}a^{7}-\frac{211337623745}{19551893516}a^{6}-\frac{322296021827}{19551893516}a^{5}-\frac{124701006037}{9775946758}a^{4}-\frac{3330933709031}{19551893516}a^{3}+\frac{4595732401751}{19551893516}a^{2}-\frac{1598933608293}{9775946758}a+\frac{202641775028}{4887973379}$, $\frac{220887431}{9775946758}a^{11}+\frac{1666501779}{9775946758}a^{10}-\frac{1518837500}{4887973379}a^{9}-\frac{3429471597}{9775946758}a^{8}+\frac{23009189723}{9775946758}a^{7}-\frac{36043939519}{9775946758}a^{6}-\frac{20640406621}{9775946758}a^{5}-\frac{8597922809}{4887973379}a^{4}+\frac{30912208286}{4887973379}a^{3}-\frac{345319442653}{9775946758}a^{2}+\frac{202044795344}{4887973379}a-\frac{62054457179}{4887973379}$, $\frac{5035770897}{19551893516}a^{11}+\frac{8997095259}{9775946758}a^{10}-\frac{598121810}{4887973379}a^{9}-\frac{66525546155}{19551893516}a^{8}+\frac{248801189989}{19551893516}a^{7}+\frac{80941041757}{19551893516}a^{6}+\frac{139851304987}{19551893516}a^{5}+\frac{49880634801}{9775946758}a^{4}+\frac{1480380730673}{19551893516}a^{3}-\frac{2170423870585}{19551893516}a^{2}+\frac{770474292745}{9775946758}a-\frac{117438007112}{4887973379}$, $\frac{113768049303}{19551893516}a^{11}+\frac{202894738407}{9775946758}a^{10}-\frac{25891601509}{9775946758}a^{9}-\frac{1501321787375}{19551893516}a^{8}+\frac{5607092854829}{19551893516}a^{7}+\frac{1859462424365}{19551893516}a^{6}+\frac{3194750644267}{19551893516}a^{5}+\frac{573803428869}{4887973379}a^{4}+\frac{33415828618437}{19551893516}a^{3}-\frac{48900299479511}{19551893516}a^{2}+\frac{17284313449639}{9775946758}a-\frac{2625138097608}{4887973379}$, $\frac{68660480683}{19551893516}a^{11}+\frac{59713834536}{4887973379}a^{10}-\frac{7139244587}{9775946758}a^{9}-\frac{893004948777}{19551893516}a^{8}+\frac{3302593119275}{19551893516}a^{7}+\frac{1362873495563}{19551893516}a^{6}+\frac{2253940122273}{19551893516}a^{5}+\frac{800973007899}{9775946758}a^{4}+\frac{20556754084989}{19551893516}a^{3}-\frac{27473128202653}{19551893516}a^{2}+\frac{9635160536455}{9775946758}a-\frac{1322966881654}{4887973379}$ 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:  \( 646762.8045308752 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 646762.8045308752 \cdot 1}{2\cdot\sqrt{311013484453206508801}}\cr\approx \mathstrut & 0.457261763090004 \end{aligned}\]

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 - 4*x^11 + 2*x^10 + 13*x^9 - 55*x^8 + 5*x^7 - 21*x^6 - 8*x^5 - 285*x^4 + 557*x^3 - 490*x^2 + 224*x - 40) 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 - 4*x^11 + 2*x^10 + 13*x^9 - 55*x^8 + 5*x^7 - 21*x^6 - 8*x^5 - 285*x^4 + 557*x^3 - 490*x^2 + 224*x - 40, 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 - 4*x^11 + 2*x^10 + 13*x^9 - 55*x^8 + 5*x^7 - 21*x^6 - 8*x^5 - 285*x^4 + 557*x^3 - 490*x^2 + 224*x - 40); 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 = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 2*x^10 + 13*x^9 - 55*x^8 + 5*x^7 - 21*x^6 - 8*x^5 - 285*x^4 + 557*x^3 - 490*x^2 + 224*x - 40); 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 12T76):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A non-solvable group of order 120
The 10 conjugacy class representatives for $C_2\times A_5$
Character table for $C_2\times A_5$

Intermediate fields

6.2.17635574401.2

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 10 sibling: data not computed
Degree 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed
Minimal sibling: 10.0.24570065271803314195279.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.3.0.1}{3} }^{4}$ ${\href{/padicField/3.6.0.1}{6} }^{2}$ ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.3.0.1}{3} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ R ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(41\) Copy content Toggle raw display 41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.5.8a1.1$x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$$5$$2$$8$$C_{10}$$$[\ ]_{5}^{2}$$
\(79\) Copy content Toggle raw display $\Q_{79}$$x + 76$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{79}$$x + 76$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{79}$$x + 76$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{79}$$x + 76$$1$$1$$0$Trivial$$[\ ]$$
79.1.2.1a1.1$x^{2} + 79$$2$$1$$1$$C_2$$$[\ ]_{2}$$
79.1.2.1a1.1$x^{2} + 79$$2$$1$$1$$C_2$$$[\ ]_{2}$$
79.1.2.1a1.1$x^{2} + 79$$2$$1$$1$$C_2$$$[\ ]_{2}$$
79.1.2.1a1.1$x^{2} + 79$$2$$1$$1$$C_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)