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Properties

Label 12.4.449...576.2
Degree $12$
Signature $[4, 4]$
Discriminant $4.492\times 10^{18}$
Root discriminant \(35.84\)
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 - 3*x^11 + 2*x^10 - 13*x^9 + 36*x^8 - 45*x^7 + 79*x^6 - 142*x^5 + 70*x^4 - 4*x^3 + 206*x^2 - 150*x - 50)
 
gp: K = bnfinit(y^12 - 3*y^11 + 2*y^10 - 13*y^9 + 36*y^8 - 45*y^7 + 79*y^6 - 142*y^5 + 70*y^4 - 4*y^3 + 206*y^2 - 150*y - 50, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 2*x^10 - 13*x^9 + 36*x^8 - 45*x^7 + 79*x^6 - 142*x^5 + 70*x^4 - 4*x^3 + 206*x^2 - 150*x - 50);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 3*x^11 + 2*x^10 - 13*x^9 + 36*x^8 - 45*x^7 + 79*x^6 - 142*x^5 + 70*x^4 - 4*x^3 + 206*x^2 - 150*x - 50)
 

\( x^{12} - 3 x^{11} + 2 x^{10} - 13 x^{9} + 36 x^{8} - 45 x^{7} + 79 x^{6} - 142 x^{5} + 70 x^{4} + \cdots - 50 \) 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:   \(4492085992834072576\) \(\medspace = 2^{17}\cdot 17^{11}\) 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:  \(35.84\)
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:  $2^{8/3}17^{11/12}\approx 85.24289022322928$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}$, $\frac{1}{75845185210}a^{11}+\frac{8291598366}{37922592605}a^{10}-\frac{22503151593}{75845185210}a^{9}+\frac{2618773156}{37922592605}a^{8}+\frac{12679667921}{75845185210}a^{7}+\frac{952348414}{7584518521}a^{6}-\frac{2242522818}{37922592605}a^{5}-\frac{10333839736}{37922592605}a^{4}+\frac{2796594977}{7584518521}a^{3}+\frac{18051248908}{37922592605}a^{2}+\frac{10254936118}{37922592605}a-\frac{209839253}{7584518521}$ 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:   $\frac{134369668}{37922592605}a^{11}-\frac{195591049}{37922592605}a^{10}-\frac{621707954}{37922592605}a^{9}-\frac{717747769}{37922592605}a^{8}+\frac{2422402723}{37922592605}a^{7}+\frac{880758988}{7584518521}a^{6}-\frac{7399216398}{37922592605}a^{5}-\frac{1616670716}{37922592605}a^{4}-\frac{5369531966}{7584518521}a^{3}+\frac{35532445113}{37922592605}a^{2}+\frac{27130216633}{37922592605}a-\frac{5882987295}{7584518521}$, $\frac{63090957}{7584518521}a^{11}-\frac{210860237}{7584518521}a^{10}+\frac{158952693}{7584518521}a^{9}-\frac{773323551}{7584518521}a^{8}+\frac{2561616938}{7584518521}a^{7}-\frac{3310624003}{7584518521}a^{6}+\frac{4691925572}{7584518521}a^{5}-\frac{10158077038}{7584518521}a^{4}+\frac{7323736710}{7584518521}a^{3}-\frac{362115781}{7584518521}a^{2}+\frac{20833384449}{7584518521}a-\frac{24564406733}{7584518521}$, $\frac{3333491241}{75845185210}a^{11}-\frac{5443400433}{75845185210}a^{10}-\frac{1193363283}{75845185210}a^{9}-\frac{41802880863}{75845185210}a^{8}+\frac{56728012511}{75845185210}a^{7}-\frac{12391674215}{15169037042}a^{6}+\frac{67876449442}{37922592605}a^{5}-\frac{105411748111}{37922592605}a^{4}-\frac{18101395756}{7584518521}a^{3}-\frac{25127900207}{37922592605}a^{2}+\frac{203470482718}{37922592605}a+\frac{9482117338}{7584518521}$, $\frac{6855087419}{75845185210}a^{11}-\frac{7013518236}{37922592605}a^{10}+\frac{1814003293}{75845185210}a^{9}-\frac{45269971131}{37922592605}a^{8}+\frac{158906668399}{75845185210}a^{7}-\frac{17010246773}{7584518521}a^{6}+\frac{205838107618}{37922592605}a^{5}-\frac{295865520344}{37922592605}a^{4}-\frac{7046987888}{7584518521}a^{3}-\frac{73723118323}{37922592605}a^{2}+\frac{552028957602}{37922592605}a+\frac{25856642075}{7584518521}$, $\frac{300761296}{37922592605}a^{11}+\frac{431535097}{37922592605}a^{10}-\frac{4157821118}{37922592605}a^{9}+\frac{1088050377}{37922592605}a^{8}-\frac{8317010959}{37922592605}a^{7}+\frac{8190275996}{7584518521}a^{6}-\frac{54955239826}{37922592605}a^{5}+\frac{92555763538}{37922592605}a^{4}-\frac{36923606474}{7584518521}a^{3}+\frac{111496985261}{37922592605}a^{2}+\frac{150317782826}{37922592605}a+\frac{8726861}{7584518521}$, $\frac{305859034}{7584518521}a^{11}-\frac{572227494}{7584518521}a^{10}+\frac{39277400}{7584518521}a^{9}-\frac{4078650289}{7584518521}a^{8}+\frac{6219066750}{7584518521}a^{7}-\frac{7573110070}{7584518521}a^{6}+\frac{17492758163}{7584518521}a^{5}-\frac{23661481447}{7584518521}a^{4}-\frac{2727610996}{7584518521}a^{3}-\frac{7182503427}{7584518521}a^{2}+\frac{53766325441}{7584518521}a+\frac{13105222187}{7584518521}$, $\frac{362170234}{37922592605}a^{11}-\frac{1717420272}{37922592605}a^{10}+\frac{2832942838}{37922592605}a^{9}-\frac{6297739972}{37922592605}a^{8}+\frac{20771363934}{37922592605}a^{7}-\frac{8382765982}{7584518521}a^{6}+\frac{61717688516}{37922592605}a^{5}-\frac{98347428948}{37922592605}a^{4}+\frac{25386537352}{7584518521}a^{3}-\frac{74686048036}{37922592605}a^{2}+\frac{2383040804}{37922592605}a-\frac{60116394439}{7584518521}$ 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:  \( 65986.6167804 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 65986.6167804 \cdot 2}{2\cdot\sqrt{4492085992834072576}}\cr\approx \mathstrut & 0.776374544503 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 2*x^10 - 13*x^9 + 36*x^8 - 45*x^7 + 79*x^6 - 142*x^5 + 70*x^4 - 4*x^3 + 206*x^2 - 150*x - 50)
 
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 - 3*x^11 + 2*x^10 - 13*x^9 + 36*x^8 - 45*x^7 + 79*x^6 - 142*x^5 + 70*x^4 - 4*x^3 + 206*x^2 - 150*x - 50, 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 - 3*x^11 + 2*x^10 - 13*x^9 + 36*x^8 - 45*x^7 + 79*x^6 - 142*x^5 + 70*x^4 - 4*x^3 + 206*x^2 - 150*x - 50);
 
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 - 3*x^11 + 2*x^10 - 13*x^9 + 36*x^8 - 45*x^7 + 79*x^6 - 142*x^5 + 70*x^4 - 4*x^3 + 206*x^2 - 150*x - 50);
 
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: 12.2.2246042996417036288.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ R ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
2.6.11.8$x^{6} + 4 x^{5} + 4 x^{2} + 4 x + 2$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 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}$