↑

Properties

Label 12.12.110...125.1
Degree $12$
Signature $[12, 0]$
Discriminant $1.108\times 10^{19}$
Root discriminant \(38.64\)
Ramified primes $3,5,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3^2:C_{12}$ (as 12T73)

Related objects

Downloads

Learn more

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 - 42*x^10 - 19*x^9 + 639*x^8 + 531*x^7 - 4122*x^6 - 4653*x^5 + 9900*x^4 + 13904*x^3 - 4356*x^2 - 11253*x - 3509)
 
Copy content gp:K = bnfinit(y^12 - 42*y^10 - 19*y^9 + 639*y^8 + 531*y^7 - 4122*y^6 - 4653*y^5 + 9900*y^4 + 13904*y^3 - 4356*y^2 - 11253*y - 3509, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 42*x^10 - 19*x^9 + 639*x^8 + 531*x^7 - 4122*x^6 - 4653*x^5 + 9900*x^4 + 13904*x^3 - 4356*x^2 - 11253*x - 3509);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 42*x^10 - 19*x^9 + 639*x^8 + 531*x^7 - 4122*x^6 - 4653*x^5 + 9900*x^4 + 13904*x^3 - 4356*x^2 - 11253*x - 3509)
 

\( x^{12} - 42 x^{10} - 19 x^{9} + 639 x^{8} + 531 x^{7} - 4122 x^{6} - 4653 x^{5} + 9900 x^{4} + \cdots - 3509 \) 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:  $[12, 0]$
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:   \(11078561287986328125\) \(\medspace = 3^{18}\cdot 5^{9}\cdot 11^{4}\) 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:  \(38.64\)
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:  $3^{97/54}5^{3/4}11^{2/3}\approx 118.99821203205568$
Ramified primes:   \(3\), \(5\), \(11\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{33}a^{9}+\frac{2}{33}a^{7}+\frac{1}{11}a^{6}+\frac{4}{11}a^{5}+\frac{14}{33}a^{4}+\frac{1}{11}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{33}a^{10}+\frac{2}{33}a^{8}+\frac{1}{11}a^{7}+\frac{1}{33}a^{6}+\frac{14}{33}a^{5}+\frac{1}{11}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{472870497}a^{11}-\frac{2952715}{472870497}a^{10}-\frac{2423141}{472870497}a^{9}-\frac{2704330}{42988227}a^{8}-\frac{9007430}{472870497}a^{7}+\frac{5389723}{157623499}a^{6}-\frac{194282939}{472870497}a^{5}+\frac{49918450}{472870497}a^{4}+\frac{32046226}{157623499}a^{3}-\frac{6398629}{14329409}a^{2}-\frac{8146100}{42988227}a+\frac{9313030}{42988227}$ 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:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
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:  $C_{2}$, which has order $2$ (assuming GRH)
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:  $11$
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:   $\frac{91725}{462239}a^{11}-\frac{267422}{1386717}a^{10}-\frac{3767616}{462239}a^{9}+\frac{1926596}{462239}a^{8}+\frac{56789486}{462239}a^{7}-\frac{20089291}{1386717}a^{6}-\frac{372047599}{462239}a^{5}-\frac{63392780}{462239}a^{4}+\frac{2914230638}{1386717}a^{3}+\frac{325816700}{462239}a^{2}-\frac{2155859662}{1386717}a-\frac{986800807}{1386717}$, $\frac{700081}{5084629}a^{11}-\frac{2096185}{15253887}a^{10}-\frac{28690711}{5084629}a^{9}+\frac{1388162}{462239}a^{8}+\frac{431667573}{5084629}a^{7}-\frac{172055399}{15253887}a^{6}-\frac{256743691}{462239}a^{5}-\frac{41260912}{462239}a^{4}+\frac{22106167114}{15253887}a^{3}+\frac{220554873}{462239}a^{2}-\frac{1488207145}{1386717}a-\frac{676833763}{1386717}$, $\frac{234357}{5084629}a^{11}-\frac{705758}{15253887}a^{10}-\frac{28748215}{15253887}a^{9}+\frac{462763}{462239}a^{8}+\frac{143935736}{5084629}a^{7}-\frac{54185986}{15253887}a^{6}-\frac{85513226}{462239}a^{5}-\frac{14814646}{462239}a^{4}+\frac{7360787483}{15253887}a^{3}+\frac{76925464}{462239}a^{2}-\frac{496764740}{1386717}a-\frac{77647019}{462239}$, $\frac{184231732}{472870497}a^{11}-\frac{184661129}{472870497}a^{10}-\frac{7546546489}{472870497}a^{9}+\frac{1344747947}{157623499}a^{8}+\frac{113510529511}{472870497}a^{7}-\frac{15140961029}{472870497}a^{6}-\frac{247521381718}{157623499}a^{5}-\frac{120015037397}{472870497}a^{4}+\frac{1937246069329}{472870497}a^{3}+\frac{58260774343}{42988227}a^{2}-\frac{43411490566}{14329409}a-\frac{59379083921}{42988227}$, $\frac{24912122}{472870497}a^{11}-\frac{7697255}{157623499}a^{10}-\frac{1028661896}{472870497}a^{9}+\frac{165105616}{157623499}a^{8}+\frac{15569800787}{472870497}a^{7}-\frac{535807136}{157623499}a^{6}-\frac{34120717270}{157623499}a^{5}-\frac{17630340736}{472870497}a^{4}+\frac{89442377128}{157623499}a^{3}+\frac{8053902449}{42988227}a^{2}-\frac{18177092789}{42988227}a-\frac{8275003061}{42988227}$, $\frac{9183695}{157623499}a^{11}-\frac{2307514}{42988227}a^{10}-\frac{1137997247}{472870497}a^{9}+\frac{536815799}{472870497}a^{8}+\frac{17240319155}{472870497}a^{7}-\frac{1567562863}{472870497}a^{6}-\frac{113503422946}{472870497}a^{5}-\frac{20934403348}{472870497}a^{4}+\frac{298003927682}{472870497}a^{3}+\frac{9148112698}{42988227}a^{2}-\frac{20159404960}{42988227}a-\frac{9235426108}{42988227}$, $\frac{60731384}{472870497}a^{11}-\frac{60094132}{472870497}a^{10}-\frac{2490808568}{472870497}a^{9}+\frac{1308248171}{472870497}a^{8}+\frac{37495794770}{472870497}a^{7}-\frac{4774331554}{472870497}a^{6}-\frac{245372667610}{472870497}a^{5}-\frac{40500545347}{472870497}a^{4}+\frac{639881695991}{472870497}a^{3}+\frac{6448150479}{14329409}a^{2}-\frac{14326889772}{14329409}a-\frac{19623925783}{42988227}$, $\frac{57319751}{472870497}a^{11}-\frac{18229179}{157623499}a^{10}-\frac{2359367278}{472870497}a^{9}+\frac{393069669}{157623499}a^{8}+\frac{35615980885}{472870497}a^{7}-\frac{1325251047}{157623499}a^{6}-\frac{77874989279}{157623499}a^{5}-\frac{40338850097}{472870497}a^{4}+\frac{203620565310}{157623499}a^{3}+\frac{18609816983}{42988227}a^{2}-\frac{41159114530}{42988227}a-\frac{18806063533}{42988227}$, $\frac{43508254}{472870497}a^{11}-\frac{17453576}{157623499}a^{10}-\frac{1747364063}{472870497}a^{9}+\frac{1204299658}{472870497}a^{8}+\frac{25883961236}{472870497}a^{7}-\frac{558160385}{42988227}a^{6}-\frac{167506344854}{472870497}a^{5}-\frac{17138578279}{472870497}a^{4}+\frac{434283537047}{472870497}a^{3}+\frac{3955796609}{14329409}a^{2}-\frac{29149109345}{42988227}a-\frac{12765446611}{42988227}$, $\frac{93957037}{472870497}a^{11}-\frac{34638205}{157623499}a^{10}-\frac{1269176985}{157623499}a^{9}+\frac{778257957}{157623499}a^{8}+\frac{56771011534}{472870497}a^{7}-\frac{3481810720}{157623499}a^{6}-\frac{122971037242}{157623499}a^{5}-\frac{50396588744}{472870497}a^{4}+\frac{319474417753}{157623499}a^{3}+\frac{28166810527}{42988227}a^{2}-\frac{64415048152}{42988227}a-\frac{9757374221}{14329409}$, $\frac{81569182}{472870497}a^{11}-\frac{26079941}{157623499}a^{10}-\frac{304351753}{42988227}a^{9}+\frac{554349584}{157623499}a^{8}+\frac{50410198222}{472870497}a^{7}-\frac{1718860164}{157623499}a^{6}-\frac{109907873916}{157623499}a^{5}-\frac{61156815815}{472870497}a^{4}+\frac{286268233357}{157623499}a^{3}+\frac{27130972687}{42988227}a^{2}-\frac{57759728617}{42988227}a-\frac{27065774024}{42988227}$ Copy content Toggle raw display (assuming GRH)
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:  \( 387641.416817 \) (assuming GRH)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 387641.416817 \cdot 1}{2\cdot\sqrt{11078561287986328125}}\cr\approx \mathstrut & 0.238516507889 \end{aligned}\] (assuming GRH)

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 - 42*x^10 - 19*x^9 + 639*x^8 + 531*x^7 - 4122*x^6 - 4653*x^5 + 9900*x^4 + 13904*x^3 - 4356*x^2 - 11253*x - 3509) 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 - 42*x^10 - 19*x^9 + 639*x^8 + 531*x^7 - 4122*x^6 - 4653*x^5 + 9900*x^4 + 13904*x^3 - 4356*x^2 - 11253*x - 3509, 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 - 42*x^10 - 19*x^9 + 639*x^8 + 531*x^7 - 4122*x^6 - 4653*x^5 + 9900*x^4 + 13904*x^3 - 4356*x^2 - 11253*x - 3509); 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 - 42*x^10 - 19*x^9 + 639*x^8 + 531*x^7 - 4122*x^6 - 4653*x^5 + 9900*x^4 + 13904*x^3 - 4356*x^2 - 11253*x - 3509); 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

$C_3^2:C_{12}$ (as 12T73):

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 108
The 18 conjugacy class representatives for $C_3^2:C_{12}$
Character table for $C_3^2:C_{12}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\)

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 18 siblings: data not computed
Degree 27 sibling: data not computed
Degree 36 siblings: 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 ${\href{/padicField/2.12.0.1}{12} }$ R R ${\href{/padicField/7.12.0.1}{12} }$ R ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.4.0.1}{4} }^{3}$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

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
\(3\) Copy content Toggle raw display 3.2.6.18a4.5$x^{12} + 15 x^{11} + 105 x^{10} + 463 x^{9} + 1440 x^{8} + 3336 x^{7} + 5912 x^{6} + 8088 x^{5} + 8496 x^{4} + 6710 x^{3} + 3810 x^{2} + 1419 x + 268$$6$$2$$18$12T73$$[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{2}$$
\(5\) 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}$$$[\ ]_{4}^{3}$$
\(11\) Copy content Toggle raw display 11.2.3.4a1.1$x^{6} + 21 x^{5} + 153 x^{4} + 427 x^{3} + 306 x^{2} + 95 x + 8$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
11.6.1.0a1.1$x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$

Spectrum of ring of integers

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