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Properties

Label 12.0.109627439316992.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.096\times 10^{14}$
Root discriminant \(14.79\)
Ramified primes $2,53$
Class number $2$
Class group [2]
Galois group $D_{12}$ (as 12T12)

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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 + 24*x^10 - 60*x^9 + 108*x^8 - 148*x^7 + 154*x^6 - 110*x^5 + 52*x^4 - 16*x^3 + 18*x^2 - 8*x + 1)
 
gp: K = bnfinit(y^12 - 6*y^11 + 24*y^10 - 60*y^9 + 108*y^8 - 148*y^7 + 154*y^6 - 110*y^5 + 52*y^4 - 16*y^3 + 18*y^2 - 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 24*x^10 - 60*x^9 + 108*x^8 - 148*x^7 + 154*x^6 - 110*x^5 + 52*x^4 - 16*x^3 + 18*x^2 - 8*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^11 + 24*x^10 - 60*x^9 + 108*x^8 - 148*x^7 + 154*x^6 - 110*x^5 + 52*x^4 - 16*x^3 + 18*x^2 - 8*x + 1)
 

\( x^{12} - 6 x^{11} + 24 x^{10} - 60 x^{9} + 108 x^{8} - 148 x^{7} + 154 x^{6} - 110 x^{5} + 52 x^{4} + \cdots + 1 \) 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:   \(109627439316992\) \(\medspace = 2^{18}\cdot 53^{5}\) 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:  \(14.79\)
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:  $2^{3/2}53^{1/2}\approx 20.591260281974$
Ramified primes:   \(2\), \(53\) 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:  \(\Q(\sqrt{53}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}+\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{13865}a^{11}-\frac{433}{13865}a^{10}-\frac{3649}{13865}a^{9}-\frac{5909}{13865}a^{8}-\frac{1145}{2773}a^{7}+\frac{4187}{13865}a^{6}+\frac{178}{2773}a^{5}+\frac{5307}{13865}a^{4}+\frac{1010}{2773}a^{3}+\frac{1028}{13865}a^{2}-\frac{804}{13865}a-\frac{6098}{13865}$ 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:   \( \frac{197524}{2773} a^{11} - \frac{1134410}{2773} a^{10} + \frac{4449295}{2773} a^{9} - \frac{10709077}{2773} a^{8} + \frac{18583600}{2773} a^{7} - \frac{24463803}{2773} a^{6} + \frac{24140990}{2773} a^{5} - \frac{15535030}{2773} a^{4} + \frac{6290123}{2773} a^{3} - \frac{1551133}{2773} a^{2} + \frac{3161634}{2773} a - \frac{769782}{2773} \)  (order $4$) 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:   $\frac{1053148}{13865}a^{11}-\frac{6049466}{13865}a^{10}+\frac{23728602}{13865}a^{9}-\frac{11424485}{2773}a^{8}+\frac{19828178}{2773}a^{7}-\frac{130535844}{13865}a^{6}+\frac{128839116}{13865}a^{5}-\frac{82943138}{13865}a^{4}+\frac{33597362}{13865}a^{3}-\frac{8278694}{13865}a^{2}+\frac{16842214}{13865}a-\frac{826440}{2773}$, $\frac{243491}{2773}a^{11}-\frac{6990356}{13865}a^{10}+\frac{27413552}{13865}a^{9}-\frac{65967279}{13865}a^{8}+\frac{22889467}{2773}a^{7}-\frac{30125278}{2773}a^{6}+\frac{148593043}{13865}a^{5}-\frac{95555662}{13865}a^{4}+\frac{38638101}{13865}a^{3}-\frac{9514059}{13865}a^{2}+\frac{19462818}{13865}a-\frac{4715483}{13865}$, $\frac{50734}{2773}a^{11}-\frac{291281}{2773}a^{10}+\frac{1142363}{2773}a^{9}-\frac{2748992}{2773}a^{8}+\frac{4769749}{2773}a^{7}-\frac{6277706}{2773}a^{6}+\frac{6192610}{2773}a^{5}-\frac{3981125}{2773}a^{4}+\frac{1609251}{2773}a^{3}-\frac{393798}{2773}a^{2}+\frac{810410}{2773}a-\frac{197524}{2773}$, $\frac{146790}{2773}a^{11}-\frac{843129}{2773}a^{10}+\frac{3306932}{2773}a^{9}-\frac{7960085}{2773}a^{8}+\frac{13813851}{2773}a^{7}-\frac{18186097}{2773}a^{6}+\frac{17948380}{2773}a^{5}-\frac{11553905}{2773}a^{4}+\frac{4680872}{2773}a^{3}-\frac{1157335}{2773}a^{2}+\frac{2351224}{2773}a-\frac{575031}{2773}$, $\frac{1010263}{13865}a^{11}-\frac{1160849}{2773}a^{10}+\frac{4553560}{2773}a^{9}-\frac{54815656}{13865}a^{8}+\frac{19028595}{2773}a^{7}-\frac{125274889}{13865}a^{6}+\frac{123653528}{13865}a^{5}-\frac{79601166}{13865}a^{4}+\frac{32221231}{13865}a^{3}-\frac{1583766}{2773}a^{2}+\frac{16143476}{13865}a-\frac{3958082}{13865}$ 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:  \( 263.773598435 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 263.773598435 \cdot 2}{4\cdot\sqrt{109627439316992}}\cr\approx \mathstrut & 0.775034265160 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 24*x^10 - 60*x^9 + 108*x^8 - 148*x^7 + 154*x^6 - 110*x^5 + 52*x^4 - 16*x^3 + 18*x^2 - 8*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 - 6*x^11 + 24*x^10 - 60*x^9 + 108*x^8 - 148*x^7 + 154*x^6 - 110*x^5 + 52*x^4 - 16*x^3 + 18*x^2 - 8*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 - 6*x^11 + 24*x^10 - 60*x^9 + 108*x^8 - 148*x^7 + 154*x^6 - 110*x^5 + 52*x^4 - 16*x^3 + 18*x^2 - 8*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 - 6*x^11 + 24*x^10 - 60*x^9 + 108*x^8 - 148*x^7 + 154*x^6 - 110*x^5 + 52*x^4 - 16*x^3 + 18*x^2 - 8*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

$D_{12}$ (as 12T12):

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 9 conjugacy class representatives for $D_{12}$
Character table for $D_{12}$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.212.1, 4.0.3392.1, 6.0.179776.1

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 12 sibling: 12.2.1452563570950144.1
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 ${\href{/padicField/3.12.0.1}{12} }$ ${\href{/padicField/5.2.0.1}{2} }^{5}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{6}$ ${\href{/padicField/11.2.0.1}{2} }^{6}$ ${\href{/padicField/13.3.0.1}{3} }^{4}$ ${\href{/padicField/17.3.0.1}{3} }^{4}$ ${\href{/padicField/19.12.0.1}{12} }$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ R ${\href{/padicField/59.2.0.1}{2} }^{6}$

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 2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.8.12.13$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 87 x^{4} + 98 x^{3} + 58 x^{2} - 2 x + 1$$4$$2$$12$$D_4$$[2, 2]^{2}$
\(53\) Copy content Toggle raw display 53.2.0.1$x^{2} + 49 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.4.2.1$x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$