Properties

Label 12.12.131...581.1
Degree $12$
Signature $[12, 0]$
Discriminant $1.310\times 10^{17}$
Root discriminant \(26.70\)
Ramified primes $3,367$
Class number $1$
Class group trivial
Galois group $S_4$ (as 12T8)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 21*x^10 + 13*x^9 + 163*x^8 - 54*x^7 - 554*x^6 + 93*x^5 + 765*x^4 - 73*x^3 - 275*x^2 - 42*x + 1)
 
gp: K = bnfinit(y^12 - y^11 - 21*y^10 + 13*y^9 + 163*y^8 - 54*y^7 - 554*y^6 + 93*y^5 + 765*y^4 - 73*y^3 - 275*y^2 - 42*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 21*x^10 + 13*x^9 + 163*x^8 - 54*x^7 - 554*x^6 + 93*x^5 + 765*x^4 - 73*x^3 - 275*x^2 - 42*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 - 21*x^10 + 13*x^9 + 163*x^8 - 54*x^7 - 554*x^6 + 93*x^5 + 765*x^4 - 73*x^3 - 275*x^2 - 42*x + 1)
 

\( x^{12} - x^{11} - 21 x^{10} + 13 x^{9} + 163 x^{8} - 54 x^{7} - 554 x^{6} + 93 x^{5} + 765 x^{4} - 73 x^{3} - 275 x^{2} - 42 x + 1 \) 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:  $[12, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(131045349590545581\) \(\medspace = 3^{9}\cdot 367^{5}\) 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:  \(26.70\)
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:  $3^{3/4}367^{1/2}\approx 43.66907302809774$
Ramified primes:   \(3\), \(367\) 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:  \(\Q(\sqrt{1101}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{12}a^{10}-\frac{1}{4}a^{9}+\frac{1}{6}a^{8}+\frac{5}{12}a^{6}+\frac{1}{6}a^{5}+\frac{1}{12}a^{4}-\frac{1}{12}a^{3}+\frac{1}{3}a^{2}-\frac{1}{6}a-\frac{5}{12}$, $\frac{1}{24}a^{11}-\frac{1}{24}a^{9}-\frac{1}{24}a^{7}-\frac{7}{24}a^{6}+\frac{7}{24}a^{5}+\frac{1}{3}a^{4}-\frac{11}{24}a^{3}-\frac{1}{3}a^{2}-\frac{11}{24}a-\frac{3}{8}$ 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

Trivial group, which has order $1$

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:  $11$
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:   $\frac{29}{24}a^{11}-\frac{11}{3}a^{10}-\frac{425}{24}a^{9}+\frac{307}{6}a^{8}+\frac{2155}{24}a^{7}-\frac{5767}{24}a^{6}-\frac{1319}{8}a^{5}+414a^{4}+\frac{435}{8}a^{3}-\frac{875}{6}a^{2}-\frac{791}{24}a+\frac{11}{24}$, $\frac{19}{24}a^{11}-\frac{5}{2}a^{10}-\frac{271}{24}a^{9}+35a^{8}+\frac{1301}{24}a^{7}-\frac{3961}{24}a^{6}-\frac{2027}{24}a^{5}+\frac{1703}{6}a^{4}-\frac{197}{24}a^{3}-\frac{277}{3}a^{2}-\frac{185}{24}a+\frac{11}{8}$, $\frac{35}{24}a^{11}-\frac{19}{4}a^{10}-\frac{485}{24}a^{9}+\frac{131}{2}a^{8}+\frac{2257}{24}a^{7}-\frac{7283}{24}a^{6}-\frac{3391}{24}a^{5}+\frac{6113}{12}a^{4}-\frac{343}{24}a^{3}-\frac{470}{3}a^{2}-\frac{565}{24}a+\frac{1}{8}$, $\frac{25}{6}a^{11}-\frac{163}{12}a^{10}-\frac{689}{12}a^{9}+\frac{1115}{6}a^{8}+\frac{1607}{6}a^{7}-\frac{10273}{12}a^{6}-415a^{5}+\frac{5767}{4}a^{4}-\frac{17}{4}a^{3}-\frac{1448}{3}a^{2}-\frac{167}{3}a+\frac{41}{12}$, $\frac{43}{24}a^{11}-\frac{17}{3}a^{10}-\frac{607}{24}a^{9}+\frac{233}{3}a^{8}+\frac{2957}{24}a^{7}-\frac{2871}{8}a^{6}-\frac{5143}{24}a^{5}+\frac{3667}{6}a^{4}+\frac{1319}{24}a^{3}-\frac{445}{2}a^{2}-\frac{303}{8}a+\frac{137}{24}$, $\frac{67}{24}a^{11}-\frac{107}{12}a^{10}-\frac{937}{24}a^{9}+\frac{733}{6}a^{8}+\frac{4469}{24}a^{7}-\frac{4505}{8}a^{6}-\frac{7279}{24}a^{5}+\frac{11381}{12}a^{4}+\frac{677}{24}a^{3}-317a^{2}-\frac{359}{8}a+\frac{59}{24}$, $\frac{11}{8}a^{11}-\frac{14}{3}a^{10}-\frac{147}{8}a^{9}+\frac{191}{3}a^{8}+\frac{653}{8}a^{7}-\frac{7031}{24}a^{6}-\frac{2681}{24}a^{5}+\frac{1480}{3}a^{4}-\frac{779}{24}a^{3}-\frac{497}{3}a^{2}-\frac{283}{24}a+\frac{23}{24}$, $\frac{13}{12}a^{11}-\frac{37}{12}a^{10}-\frac{49}{3}a^{9}+\frac{251}{6}a^{8}+\frac{1073}{12}a^{7}-\frac{381}{2}a^{6}-\frac{2425}{12}a^{5}+\frac{3835}{12}a^{4}+\frac{985}{6}a^{3}-\frac{231}{2}a^{2}-\frac{293}{4}a-\frac{25}{3}$, $\frac{3}{4}a^{11}-\frac{29}{12}a^{10}-\frac{21}{2}a^{9}+\frac{199}{6}a^{8}+\frac{203}{4}a^{7}-\frac{923}{6}a^{6}-\frac{1051}{12}a^{5}+\frac{3169}{12}a^{4}+\frac{127}{6}a^{3}-\frac{302}{3}a^{2}-\frac{185}{12}a+\frac{17}{6}$, $\frac{5}{3}a^{11}-\frac{59}{12}a^{10}-\frac{299}{12}a^{9}+\frac{206}{3}a^{8}+\frac{785}{6}a^{7}-\frac{1293}{4}a^{6}-\frac{782}{3}a^{5}+\frac{6749}{12}a^{4}+\frac{1579}{12}a^{3}-211a^{2}-60a+\frac{7}{12}$, $\frac{9}{4}a^{11}-\frac{41}{6}a^{10}-\frac{131}{4}a^{9}+\frac{563}{6}a^{8}+\frac{673}{4}a^{7}-\frac{5219}{12}a^{6}-\frac{3977}{12}a^{5}+\frac{4489}{6}a^{4}+\frac{2077}{12}a^{3}-\frac{865}{3}a^{2}-\frac{955}{12}a+\frac{23}{12}$ 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:  \( 50889.5043205 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 50889.5043205 \cdot 1}{2\cdot\sqrt{131045349590545581}}\cr\approx \mathstrut & 0.287903779490 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 21*x^10 + 13*x^9 + 163*x^8 - 54*x^7 - 554*x^6 + 93*x^5 + 765*x^4 - 73*x^3 - 275*x^2 - 42*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 - x^11 - 21*x^10 + 13*x^9 + 163*x^8 - 54*x^7 - 554*x^6 + 93*x^5 + 765*x^4 - 73*x^3 - 275*x^2 - 42*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 - x^11 - 21*x^10 + 13*x^9 + 163*x^8 - 54*x^7 - 554*x^6 + 93*x^5 + 765*x^4 - 73*x^3 - 275*x^2 - 42*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 - x^11 - 21*x^10 + 13*x^9 + 163*x^8 - 54*x^7 - 554*x^6 + 93*x^5 + 765*x^4 - 73*x^3 - 275*x^2 - 42*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 12T8):

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

3.3.1101.1, 4.4.9909.1 x2, 6.6.10909809.1

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: 4.4.9909.1
Degree 6 siblings: 6.6.1334633301.2, 6.6.10909809.1
Degree 8 sibling: 8.8.13224881379609.1
Degree 12 sibling: deg 12
Minimal sibling: 4.4.9909.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{3}$ R ${\href{/padicField/5.3.0.1}{3} }^{4}$ ${\href{/padicField/7.3.0.1}{3} }^{4}$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.3.0.1}{3} }^{4}$ ${\href{/padicField/17.3.0.1}{3} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{3}$ ${\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{3}$

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
\(3\) Copy content Toggle raw display 3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
\(367\) Copy content Toggle raw display $\Q_{367}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{367}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$