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Properties

Label 12.0.648669553496064.1
Degree $12$
Signature $[0, 6]$
Discriminant $6.487\times 10^{14}$
Root discriminant \(17.15\)
Ramified primes $2,3,17$
Class number $2$
Class group [2]
Galois group $S_4$ (as 12T9)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 13*x^10 - 28*x^9 + 62*x^8 - 112*x^7 + 187*x^6 - 236*x^5 + 218*x^4 - 134*x^3 + 43*x^2 - 2*x + 1)
 
gp: K = bnfinit(y^12 - 4*y^11 + 13*y^10 - 28*y^9 + 62*y^8 - 112*y^7 + 187*y^6 - 236*y^5 + 218*y^4 - 134*y^3 + 43*y^2 - 2*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 13*x^10 - 28*x^9 + 62*x^8 - 112*x^7 + 187*x^6 - 236*x^5 + 218*x^4 - 134*x^3 + 43*x^2 - 2*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 + 13*x^10 - 28*x^9 + 62*x^8 - 112*x^7 + 187*x^6 - 236*x^5 + 218*x^4 - 134*x^3 + 43*x^2 - 2*x + 1)
 

\( x^{12} - 4 x^{11} + 13 x^{10} - 28 x^{9} + 62 x^{8} - 112 x^{7} + 187 x^{6} - 236 x^{5} + 218 x^{4} + \cdots + 1 \) 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:   \(648669553496064\) \(\medspace = 2^{12}\cdot 3^{8}\cdot 17^{6}\) 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:  \(17.15\)
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:  $2^{7/6}3^{3/4}17^{1/2}\approx 21.09925220183081$
Ramified primes:   \(2\), \(3\), \(17\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{4625007}a^{11}+\frac{641353}{4625007}a^{10}-\frac{1094863}{4625007}a^{9}-\frac{194668}{4625007}a^{8}-\frac{1162118}{4625007}a^{7}-\frac{302836}{4625007}a^{6}-\frac{1902638}{4625007}a^{5}-\frac{644777}{4625007}a^{4}+\frac{141794}{1541669}a^{3}-\frac{424338}{1541669}a^{2}+\frac{670762}{4625007}a-\frac{582360}{1541669}$ 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

$C_{2}$, 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:  $5$
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:   $\frac{474968}{4625007}a^{11}-\frac{2392682}{4625007}a^{10}+\frac{2604786}{1541669}a^{9}-\frac{17972377}{4625007}a^{8}+\frac{12685526}{1541669}a^{7}-\frac{24049997}{1541669}a^{6}+\frac{40178560}{1541669}a^{5}-\frac{159312200}{4625007}a^{4}+\frac{148948874}{4625007}a^{3}-\frac{27707849}{1541669}a^{2}+\frac{20003456}{4625007}a-\frac{670852}{4625007}$, $\frac{357299}{4625007}a^{11}-\frac{659317}{1541669}a^{10}+\frac{6595382}{4625007}a^{9}-\frac{5386606}{1541669}a^{8}+\frac{34195882}{4625007}a^{7}-\frac{67252966}{4625007}a^{6}+\frac{113165977}{4625007}a^{5}-\frac{53581851}{1541669}a^{4}+\frac{161916760}{4625007}a^{3}-\frac{38646982}{1541669}a^{2}+\frac{45604175}{4625007}a-\frac{8158820}{4625007}$, $\frac{57634}{159483}a^{11}-\frac{220879}{159483}a^{10}+\frac{239899}{53161}a^{9}-\frac{1514453}{159483}a^{8}+\frac{1128077}{53161}a^{7}-\frac{1999060}{53161}a^{6}+\frac{3346328}{53161}a^{5}-\frac{12330301}{159483}a^{4}+\frac{11158984}{159483}a^{3}-\frac{2129970}{53161}a^{2}+\frac{1772221}{159483}a+\frac{31321}{159483}$, $a$, $\frac{688313}{4625007}a^{11}-\frac{742441}{1541669}a^{10}+\frac{7162820}{4625007}a^{9}-\frac{4590213}{1541669}a^{8}+\frac{32575441}{4625007}a^{7}-\frac{53931931}{4625007}a^{6}+\frac{91304221}{4625007}a^{5}-\frac{33545437}{1541669}a^{4}+\frac{91600888}{4625007}a^{3}-\frac{17419758}{1541669}a^{2}+\frac{30630773}{4625007}a-\frac{597653}{4625007}$ 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:  \( 312.005870596 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 312.005870596 \cdot 2}{2\cdot\sqrt{648669553496064}}\cr\approx \mathstrut & 0.753755022133 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 13*x^10 - 28*x^9 + 62*x^8 - 112*x^7 + 187*x^6 - 236*x^5 + 218*x^4 - 134*x^3 + 43*x^2 - 2*x + 1)
 
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 - 4*x^11 + 13*x^10 - 28*x^9 + 62*x^8 - 112*x^7 + 187*x^6 - 236*x^5 + 218*x^4 - 134*x^3 + 43*x^2 - 2*x + 1, 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 - 4*x^11 + 13*x^10 - 28*x^9 + 62*x^8 - 112*x^7 + 187*x^6 - 236*x^5 + 218*x^4 - 134*x^3 + 43*x^2 - 2*x + 1);
 
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 - 4*x^11 + 13*x^10 - 28*x^9 + 62*x^8 - 112*x^7 + 187*x^6 - 236*x^5 + 218*x^4 - 134*x^3 + 43*x^2 - 2*x + 1);
 
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

$S_4$ (as 12T9):

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 24
The 5 conjugacy class representatives for $S_4$
Character table for $S_4$

Intermediate fields

\(\Q(\sqrt{-51}) \), 3.1.204.1 x3, 6.0.8489664.1, 6.2.1498176.1, 6.0.2122416.1

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

Galois closure: deg 24
Degree 4 sibling: 4.2.7344.1
Degree 6 siblings: 6.2.1498176.1, 6.0.8489664.1
Degree 8 sibling: 8.0.15587023104.6
Degree 12 sibling: 12.2.457884390703104.1
Minimal sibling: 4.2.7344.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.3.0.1}{3} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.3.0.1}{3} }^{4}$ R ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.3.0.1}{3} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{6}$

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 2.12.12.28$x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 130 x^{7} + 159 x^{6} + 132 x^{5} + 10 x^{4} - 100 x^{3} - 53 x^{2} + 22 x + 19$$6$$2$$12$$S_4$$[4/3, 4/3]_{3}^{2}$
\(3\) Copy content Toggle raw display 3.4.3.2$x^{4} + 6$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} + 6$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(17\) Copy content Toggle raw display 17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$