Properties

Label 12.0.218...776.81
Degree $12$
Signature $[0, 6]$
Discriminant $2.190\times 10^{17}$
Root discriminant \(27.86\)
Ramified primes $2,13$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $S_5$ (as 12T74)

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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 + 13*x^10 - 2*x^9 - 36*x^8 + 30*x^7 + 187*x^6 - 726*x^5 + 1395*x^4 - 1700*x^3 + 1378*x^2 - 716*x + 194)
 
gp: K = bnfinit(y^12 - 6*y^11 + 13*y^10 - 2*y^9 - 36*y^8 + 30*y^7 + 187*y^6 - 726*y^5 + 1395*y^4 - 1700*y^3 + 1378*y^2 - 716*y + 194, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 13*x^10 - 2*x^9 - 36*x^8 + 30*x^7 + 187*x^6 - 726*x^5 + 1395*x^4 - 1700*x^3 + 1378*x^2 - 716*x + 194);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^11 + 13*x^10 - 2*x^9 - 36*x^8 + 30*x^7 + 187*x^6 - 726*x^5 + 1395*x^4 - 1700*x^3 + 1378*x^2 - 716*x + 194)
 

\( x^{12} - 6 x^{11} + 13 x^{10} - 2 x^{9} - 36 x^{8} + 30 x^{7} + 187 x^{6} - 726 x^{5} + 1395 x^{4} + \cdots + 194 \) 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:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(218971048064843776\) \(\medspace = 2^{28}\cdot 13^{8}\) 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:  \(27.86\)
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:  $2^{11/4}13^{2/3}\approx 37.1930153725407$
Ramified primes:   \(2\), \(13\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{509500580}a^{11}+\frac{99686091}{509500580}a^{10}-\frac{6261375}{50950058}a^{9}+\frac{14337842}{127375145}a^{8}+\frac{140938}{25475029}a^{7}+\frac{8440801}{50950058}a^{6}-\frac{201152643}{509500580}a^{5}-\frac{8178017}{509500580}a^{4}+\frac{53465079}{127375145}a^{3}-\frac{23391062}{127375145}a^{2}-\frac{14361329}{254750290}a-\frac{45525051}{254750290}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

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:  $5$
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:   $\frac{86936551}{509500580}a^{11}-\frac{466315259}{509500580}a^{10}+\frac{73495269}{50950058}a^{9}+\frac{190180267}{127375145}a^{8}-\frac{162973408}{25475029}a^{7}-\frac{44884435}{50950058}a^{6}+\frac{18921825967}{509500580}a^{5}-\frac{49695210867}{509500580}a^{4}+\frac{17546501149}{127375145}a^{3}-\frac{14062479927}{127375145}a^{2}+\frac{11091008651}{254750290}a-\frac{1366861571}{254750290}$, $\frac{373469}{50950058}a^{11}-\frac{1211443}{50950058}a^{10}-\frac{614390}{25475029}a^{9}+\frac{6111457}{25475029}a^{8}-\frac{5309978}{25475029}a^{7}-\frac{22399535}{25475029}a^{6}+\frac{106540751}{50950058}a^{5}+\frac{16345895}{50950058}a^{4}-\frac{142207965}{25475029}a^{3}+\frac{222721320}{25475029}a^{2}-\frac{176899844}{25475029}a+\frac{68357942}{25475029}$, $\frac{75791511}{101900116}a^{11}-\frac{386018891}{101900116}a^{10}+\frac{299680455}{50950058}a^{9}+\frac{129908234}{25475029}a^{8}-\frac{585238853}{25475029}a^{7}-\frac{115457639}{50950058}a^{6}+\frac{14475202771}{101900116}a^{5}-\frac{40960171179}{101900116}a^{4}+\frac{15800636834}{25475029}a^{3}-\frac{15138365177}{25475029}a^{2}+\frac{18393820425}{50950058}a-\frac{6049954167}{50950058}$, $\frac{2088532}{127375145}a^{11}-\frac{13122423}{127375145}a^{10}+\frac{3695111}{25475029}a^{9}+\frac{20082691}{127375145}a^{8}-\frac{16273287}{25475029}a^{7}-\frac{5323113}{25475029}a^{6}+\frac{448580159}{127375145}a^{5}-\frac{1336009154}{127375145}a^{4}+\frac{1831165707}{127375145}a^{3}-\frac{1767429941}{127375145}a^{2}+\frac{917431724}{127375145}a-\frac{277532009}{127375145}$, $\frac{24283298}{127375145}a^{11}-\frac{130326487}{127375145}a^{10}+\frac{44316557}{25475029}a^{9}+\frac{138091759}{127375145}a^{8}-\frac{166120098}{25475029}a^{7}+\frac{15269050}{25475029}a^{6}+\frac{4836085746}{127375145}a^{5}-\frac{14302433401}{127375145}a^{4}+\frac{22766008328}{127375145}a^{3}-\frac{22644674594}{127375145}a^{2}+\frac{14221441156}{127375145}a-\frac{5108995591}{127375145}$ Copy content Toggle raw display (assuming GRH)
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:  \( 16151.7410422 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 16151.7410422 \cdot 3}{2\cdot\sqrt{218971048064843776}}\cr\approx \mathstrut & 3.18563829707 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 13*x^10 - 2*x^9 - 36*x^8 + 30*x^7 + 187*x^6 - 726*x^5 + 1395*x^4 - 1700*x^3 + 1378*x^2 - 716*x + 194)
 
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 + 13*x^10 - 2*x^9 - 36*x^8 + 30*x^7 + 187*x^6 - 726*x^5 + 1395*x^4 - 1700*x^3 + 1378*x^2 - 716*x + 194, 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 + 13*x^10 - 2*x^9 - 36*x^8 + 30*x^7 + 187*x^6 - 726*x^5 + 1395*x^4 - 1700*x^3 + 1378*x^2 - 716*x + 194);
 
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 + 13*x^10 - 2*x^9 - 36*x^8 + 30*x^7 + 187*x^6 - 726*x^5 + 1395*x^4 - 1700*x^3 + 1378*x^2 - 716*x + 194);
 
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

$S_5$ (as 12T74):

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 non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

\(\Q(\sqrt{-2}) \), 6.0.58492928.4

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 5 sibling: 5.3.346112.1
Degree 6 sibling: 6.0.58492928.4
Degree 10 siblings: data not computed
Degree 15 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: 5.3.346112.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ R ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.4.11.17$x^{4} + 8 x^{3} + 8 x + 2$$4$$1$$11$$D_{4}$$[3, 4]^{2}$
2.4.11.17$x^{4} + 8 x^{3} + 8 x + 2$$4$$1$$11$$D_{4}$$[3, 4]^{2}$
2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
\(13\) Copy content Toggle raw display 13.6.4.2$x^{6} - 156 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.2$x^{6} - 156 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$