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Properties

Label 12.6.179...304.1
Degree $12$
Signature $[6, 3]$
Discriminant $-1.797\times 10^{19}$
Root discriminant \(40.23\)
Ramified primes $2,17$
Class number $2$
Class group [2]
Galois group $S_4^2:C_4$ (as 12T238)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 - 9*x^10 + 66*x^9 + 100*x^8 - 250*x^7 - 520*x^6 + 218*x^5 + 733*x^4 - 620*x^3 - 2015*x^2 - 1472*x - 358)
 
gp: K = bnfinit(y^12 - 6*y^11 - 9*y^10 + 66*y^9 + 100*y^8 - 250*y^7 - 520*y^6 + 218*y^5 + 733*y^4 - 620*y^3 - 2015*y^2 - 1472*y - 358, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 - 9*x^10 + 66*x^9 + 100*x^8 - 250*x^7 - 520*x^6 + 218*x^5 + 733*x^4 - 620*x^3 - 2015*x^2 - 1472*x - 358);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^11 - 9*x^10 + 66*x^9 + 100*x^8 - 250*x^7 - 520*x^6 + 218*x^5 + 733*x^4 - 620*x^3 - 2015*x^2 - 1472*x - 358)
 

\( x^{12} - 6 x^{11} - 9 x^{10} + 66 x^{9} + 100 x^{8} - 250 x^{7} - 520 x^{6} + 218 x^{5} + 733 x^{4} + \cdots - 358 \) Copy content Toggle raw display

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

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[6, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-17968343971336290304\) \(\medspace = -\,2^{19}\cdot 17^{11}\) 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:  \(40.23\)
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:  $2^{55/24}17^{11/12}\approx 65.73125404566245$
Ramified primes:   \(2\), \(17\) 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:  \(\Q(\sqrt{-34}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{324410234}a^{11}-\frac{20025653}{324410234}a^{10}-\frac{27656649}{324410234}a^{9}+\frac{55105617}{324410234}a^{8}-\frac{103016275}{324410234}a^{7}+\frac{58247787}{324410234}a^{6}+\frac{140786925}{324410234}a^{5}-\frac{161333563}{324410234}a^{4}+\frac{24806}{162205117}a^{3}+\frac{78073579}{162205117}a^{2}+\frac{21913611}{162205117}a+\frac{26893023}{162205117}$ 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:  $2$

Class group and class number

$C_{2}$, which has order $2$

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:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{457051057}{162205117}a^{11}-\frac{3086575082}{162205117}a^{10}-\frac{3588441469}{324410234}a^{9}+\frac{63121345129}{324410234}a^{8}+\frac{43832034375}{324410234}a^{7}-\frac{262325592095}{324410234}a^{6}-\frac{277563407747}{324410234}a^{5}+\frac{411620307913}{324410234}a^{4}+\frac{360411803063}{324410234}a^{3}-\frac{844248903895}{324410234}a^{2}-\frac{602392185735}{162205117}a-\frac{213866668928}{162205117}$, $\frac{3376875571}{324410234}a^{11}-\frac{22865214181}{324410234}a^{10}-\frac{6380962766}{162205117}a^{9}+\frac{116363938964}{162205117}a^{8}+\frac{79111588295}{162205117}a^{7}-\frac{483165881471}{162205117}a^{6}-\frac{505411608066}{162205117}a^{5}+\frac{757966332626}{162205117}a^{4}+\frac{1306557952437}{324410234}a^{3}-\frac{3101203198585}{324410234}a^{2}-\frac{2206526227316}{162205117}a-\frac{783852831822}{162205117}$, $\frac{692114253}{324410234}a^{11}-\frac{2334524999}{162205117}a^{10}-\frac{2744391247}{324410234}a^{9}+\frac{47719595873}{324410234}a^{8}+\frac{33599475571}{324410234}a^{7}-\frac{197992012205}{324410234}a^{6}-\frac{212177783313}{324410234}a^{5}+\frac{308744692533}{324410234}a^{4}+\frac{138508720971}{162205117}a^{3}-\frac{635120763815}{324410234}a^{2}-\frac{460467991998}{162205117}a-\frac{166372873488}{162205117}$, $\frac{2924879}{324410234}a^{11}-\frac{19762053}{324410234}a^{10}-\frac{11202103}{324410234}a^{9}+\frac{200976889}{324410234}a^{8}+\frac{139472071}{324410234}a^{7}-\frac{847135603}{324410234}a^{6}-\frac{868377549}{324410234}a^{5}+\frac{1434219247}{324410234}a^{4}+\frac{535476526}{162205117}a^{3}-\frac{1295989052}{162205117}a^{2}-\frac{1664585183}{162205117}a-\frac{586808529}{162205117}$, $\frac{528664239}{324410234}a^{11}-\frac{1805761818}{162205117}a^{10}-\frac{874962406}{162205117}a^{9}+\frac{18154631788}{162205117}a^{8}+\frac{11332581173}{162205117}a^{7}-\frac{75311748533}{162205117}a^{6}-\frac{74781489752}{162205117}a^{5}+\frac{119009164598}{162205117}a^{4}+\frac{188491767673}{324410234}a^{3}-\frac{240690018737}{162205117}a^{2}-\frac{331884734873}{162205117}a-\frac{116321202283}{162205117}$, $\frac{151537332}{162205117}a^{11}-\frac{2046594799}{324410234}a^{10}-\frac{1188173957}{324410234}a^{9}+\frac{20903063333}{324410234}a^{8}+\frac{14579514465}{324410234}a^{7}-\frac{86765444687}{324410234}a^{6}-\frac{92406198277}{324410234}a^{5}+\frac{135534509795}{324410234}a^{4}+\frac{120568696633}{324410234}a^{3}-\frac{138957324009}{162205117}a^{2}-\frac{200486547550}{162205117}a-\frac{71938923821}{162205117}$, $\frac{2000418555}{324410234}a^{11}-\frac{6764897726}{162205117}a^{10}-\frac{7672101975}{324410234}a^{9}+\frac{137874498399}{324410234}a^{8}+\frac{94760536077}{324410234}a^{7}-\frac{572346018769}{324410234}a^{6}-\frac{603117169811}{324410234}a^{5}+\frac{896246533439}{324410234}a^{4}+\frac{390858794192}{162205117}a^{3}-\frac{1836458045061}{324410234}a^{2}-\frac{1314152803460}{162205117}a-\frac{469437126542}{162205117}$, $\frac{4617620241}{324410234}a^{11}-\frac{31284747049}{324410234}a^{10}-\frac{17357974141}{324410234}a^{9}+\frac{318598051417}{324410234}a^{8}+\frac{214541565589}{324410234}a^{7}-\frac{1323732302427}{324410234}a^{6}-\frac{1373454199623}{324410234}a^{5}+\frac{2083370880575}{324410234}a^{4}+\frac{883993703772}{162205117}a^{3}-\frac{2127444800995}{162205117}a^{2}-\frac{2997795034708}{162205117}a-\frac{1057746562621}{162205117}$ 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:  \( 389304.384241 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 389304.384241 \cdot 2}{2\cdot\sqrt{17968343971336290304}}\cr\approx \mathstrut & 1.45799101403 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 - 9*x^10 + 66*x^9 + 100*x^8 - 250*x^7 - 520*x^6 + 218*x^5 + 733*x^4 - 620*x^3 - 2015*x^2 - 1472*x - 358)
 
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 - 6*x^11 - 9*x^10 + 66*x^9 + 100*x^8 - 250*x^7 - 520*x^6 + 218*x^5 + 733*x^4 - 620*x^3 - 2015*x^2 - 1472*x - 358, 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 - 6*x^11 - 9*x^10 + 66*x^9 + 100*x^8 - 250*x^7 - 520*x^6 + 218*x^5 + 733*x^4 - 620*x^3 - 2015*x^2 - 1472*x - 358);
 
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 - 6*x^11 - 9*x^10 + 66*x^9 + 100*x^8 - 250*x^7 - 520*x^6 + 218*x^5 + 733*x^4 - 620*x^3 - 2015*x^2 - 1472*x - 358);
 
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

$S_4^2:C_4$ (as 12T238):

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 2304
The 40 conjugacy class representatives for $S_4^2:C_4$
Character table for $S_4^2:C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 6.4.45435424.1

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: 12.4.8984171985668145152.5

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.12.0.1}{12} }$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ R ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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
\(2\) Copy content Toggle raw display 2.2.3.1$x^{2} + 4 x + 2$$2$$1$$3$$C_2$$[3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.6.8.2$x^{6} + 2 x^{4} + 2 x^{3} + 2$$6$$1$$8$$S_4\times C_2$$[4/3, 4/3, 2]_{3}^{2}$
\(17\) Copy content Toggle raw display 17.12.11.2$x^{12} + 34$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$