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Properties

Label 12.4.112...144.1
Degree $12$
Signature $[4, 4]$
Discriminant $1.123\times 10^{18}$
Root discriminant \(31.93\)
Ramified primes $2,17$
Class number $2$
Class group [2]
Galois group $S_4^2:D_4$ (as 12T260)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 68*x^6 - 663*x^4 - 272*x^2 + 34)
 
gp: K = bnfinit(y^12 + 68*y^6 - 663*y^4 - 272*y^2 + 34, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 68*x^6 - 663*x^4 - 272*x^2 + 34);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 + 68*x^6 - 663*x^4 - 272*x^2 + 34)
 

\( x^{12} + 68x^{6} - 663x^{4} - 272x^{2} + 34 \) 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:  $[4, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1123021498208518144\) \(\medspace = 2^{15}\cdot 17^{11}\) 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:  \(31.93\)
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:  $2^{37/12}17^{11/12}\approx 113.78560715460762$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{6195916}a^{10}+\frac{174131}{1548979}a^{8}+\frac{812839}{6195916}a^{6}-\frac{1}{2}a^{5}-\frac{586157}{1548979}a^{4}-\frac{1331507}{3097958}a^{2}-\frac{94609}{1548979}$, $\frac{1}{12391832}a^{11}-\frac{1}{12391832}a^{10}-\frac{852455}{12391832}a^{9}+\frac{852455}{12391832}a^{8}-\frac{2285119}{12391832}a^{7}+\frac{2285119}{12391832}a^{6}+\frac{2302309}{12391832}a^{5}-\frac{2302309}{12391832}a^{4}+\frac{1766451}{6195916}a^{3}-\frac{1766451}{6195916}a^{2}-\frac{1738197}{6195916}a+\frac{1738197}{6195916}$ 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:  $7$
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:   $\frac{39422}{1548979}a^{11}+\frac{54387}{6195916}a^{10}+\frac{22559}{6195916}a^{9}+\frac{5091}{1548979}a^{8}+\frac{10485}{1548979}a^{7}+\frac{14033}{6195916}a^{6}+\frac{10744383}{6195916}a^{5}+\frac{1981059}{3097958}a^{4}-\frac{25618826}{1548979}a^{3}-\frac{17392749}{3097958}a^{2}-\frac{27307129}{3097958}a-\frac{4438382}{1548979}$, $\frac{39422}{1548979}a^{11}-\frac{54387}{6195916}a^{10}+\frac{22559}{6195916}a^{9}-\frac{5091}{1548979}a^{8}+\frac{10485}{1548979}a^{7}-\frac{14033}{6195916}a^{6}+\frac{10744383}{6195916}a^{5}-\frac{1981059}{3097958}a^{4}-\frac{25618826}{1548979}a^{3}+\frac{17392749}{3097958}a^{2}-\frac{27307129}{3097958}a+\frac{4438382}{1548979}$, $\frac{488511}{12391832}a^{11}-\frac{34877}{12391832}a^{10}+\frac{65771}{12391832}a^{9}-\frac{29891}{12391832}a^{8}+\frac{26079}{12391832}a^{7}+\frac{27855}{12391832}a^{6}+\frac{32991411}{12391832}a^{5}-\frac{1657591}{12391832}a^{4}-\frac{160354339}{6195916}a^{3}+\frac{9873093}{6195916}a^{2}-\frac{83397439}{6195916}a+\frac{5352583}{6195916}$, $\frac{488511}{12391832}a^{11}+\frac{34877}{12391832}a^{10}+\frac{65771}{12391832}a^{9}+\frac{29891}{12391832}a^{8}+\frac{26079}{12391832}a^{7}-\frac{27855}{12391832}a^{6}+\frac{32991411}{12391832}a^{5}+\frac{1657591}{12391832}a^{4}-\frac{160354339}{6195916}a^{3}-\frac{9873093}{6195916}a^{2}-\frac{83397439}{6195916}a-\frac{5352583}{6195916}$, $\frac{143651}{12391832}a^{11}-\frac{7817}{12391832}a^{10}+\frac{70619}{12391832}a^{9}-\frac{66923}{12391832}a^{8}+\frac{211}{12391832}a^{7}-\frac{50605}{12391832}a^{6}+\frac{9581827}{12391832}a^{5}-\frac{1111431}{12391832}a^{4}-\frac{44658591}{6195916}a^{3}-\frac{3846619}{6195916}a^{2}-\frac{35497943}{6195916}a+\frac{2940119}{6195916}$, $\frac{584075}{12391832}a^{11}+\frac{97123}{12391832}a^{10}-\frac{40281}{12391832}a^{9}-\frac{59415}{12391832}a^{8}-\frac{26617}{12391832}a^{7}+\frac{98483}{12391832}a^{6}+\frac{40067443}{12391832}a^{5}+\frac{5842441}{12391832}a^{4}-\frac{196137891}{6195916}a^{3}-\frac{32873147}{6195916}a^{2}-\frac{61359839}{6195916}a-\frac{1881937}{6195916}$, $\frac{871377}{12391832}a^{11}-\frac{314671}{12391832}a^{10}+\frac{100957}{12391832}a^{9}-\frac{22041}{12391832}a^{8}-\frac{26195}{12391832}a^{7}-\frac{154615}{12391832}a^{6}+\frac{59231097}{12391832}a^{5}-\frac{21690913}{12391832}a^{4}-\frac{285455073}{6195916}a^{3}+\frac{102444143}{6195916}a^{2}-\frac{157139389}{6195916}a+\frac{49780825}{6195916}$ 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:  \( 96240.6144719 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 96240.6144719 \cdot 2}{2\cdot\sqrt{1123021498208518144}}\cr\approx \mathstrut & 2.26466416583 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 + 68*x^6 - 663*x^4 - 272*x^2 + 34)
 
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 + 68*x^6 - 663*x^4 - 272*x^2 + 34, 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 + 68*x^6 - 663*x^4 - 272*x^2 + 34);
 
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 + 68*x^6 - 663*x^4 - 272*x^2 + 34);
 
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

$S_4^2:D_4$ (as 12T260):

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 4608
The 65 conjugacy class representatives for $S_4^2:D_4$
Character table for $S_4^2:D_4$

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: 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 R ${\href{/padicField/3.6.0.1}{6} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ R ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.4.11.12$x^{4} + 26$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{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}$