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Properties

Label 12.0.177...125.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.773\times 10^{21}$
Root discriminant \(58.98\)
Ramified primes $3,5,17$
Class number $12$ (GRH)
Class group [12] (GRH)
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 - 3*x^11 - 3*x^10 + 18*x^9 - 39*x^8 - 216*x^7 + 78*x^6 + 1647*x^5 + 2997*x^4 + 729*x^3 - 3645*x^2 + 6561)
 
gp: K = bnfinit(y^12 - 3*y^11 - 3*y^10 + 18*y^9 - 39*y^8 - 216*y^7 + 78*y^6 + 1647*y^5 + 2997*y^4 + 729*y^3 - 3645*y^2 + 6561, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 - 3*x^10 + 18*x^9 - 39*x^8 - 216*x^7 + 78*x^6 + 1647*x^5 + 2997*x^4 + 729*x^3 - 3645*x^2 + 6561);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 3*x^11 - 3*x^10 + 18*x^9 - 39*x^8 - 216*x^7 + 78*x^6 + 1647*x^5 + 2997*x^4 + 729*x^3 - 3645*x^2 + 6561)
 

\( x^{12} - 3 x^{11} - 3 x^{10} + 18 x^{9} - 39 x^{8} - 216 x^{7} + 78 x^{6} + 1647 x^{5} + 2997 x^{4} + \cdots + 6561 \) 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:   \(1772506678315561228125\) \(\medspace = 3^{14}\cdot 5^{5}\cdot 17^{9}\) 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:  \(58.98\)
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:  $3^{7/6}5^{1/2}17^{3/4}\approx 67.44708071823815$
Ramified primes:   \(3\), \(5\), \(17\) 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:  \(\Q(\sqrt{85}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{45}a^{7}-\frac{2}{15}a^{6}+\frac{2}{15}a^{5}+\frac{1}{5}a^{4}-\frac{7}{15}a^{3}-\frac{1}{5}a^{2}-\frac{7}{15}a-\frac{1}{5}$, $\frac{1}{135}a^{8}+\frac{1}{9}a^{6}-\frac{1}{3}a^{5}-\frac{4}{45}a^{4}+\frac{1}{3}a^{3}-\frac{2}{9}a^{2}+\frac{1}{3}a-\frac{2}{5}$, $\frac{1}{2025}a^{9}+\frac{2}{675}a^{8}+\frac{2}{675}a^{7}-\frac{19}{225}a^{6}-\frac{157}{675}a^{5}+\frac{12}{25}a^{4}-\frac{172}{675}a^{3}+\frac{11}{25}a^{2}-\frac{2}{15}a-\frac{8}{25}$, $\frac{1}{6075}a^{10}+\frac{1}{405}a^{8}+\frac{7}{675}a^{7}-\frac{26}{405}a^{6}-\frac{298}{675}a^{5}-\frac{136}{2025}a^{4}-\frac{187}{675}a^{3}+\frac{34}{75}a^{2}-\frac{8}{75}a-\frac{4}{25}$, $\frac{1}{117642375}a^{11}+\frac{1052}{13071375}a^{10}-\frac{1817}{7842825}a^{9}-\frac{21143}{13071375}a^{8}-\frac{33772}{39214125}a^{7}+\frac{392762}{13071375}a^{6}-\frac{15316993}{39214125}a^{5}+\frac{3518897}{13071375}a^{4}+\frac{98954}{290475}a^{3}-\frac{182581}{1452375}a^{2}-\frac{222022}{484125}a+\frac{37906}{161375}$ 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:  $5$

Class group and class number

$C_{12}$, which has order $12$ (assuming GRH)

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:   $\frac{44152}{117642375}a^{11}-\frac{77833}{39214125}a^{10}+\frac{24376}{7842825}a^{9}+\frac{16364}{13071375}a^{8}-\frac{848404}{39214125}a^{7}-\frac{129292}{4357125}a^{6}+\frac{4650554}{39214125}a^{5}+\frac{1442588}{4357125}a^{4}+\frac{87104}{290475}a^{3}-\frac{363952}{1452375}a^{2}-\frac{111028}{161375}a+\frac{141142}{161375}$, $\frac{1414366}{117642375}a^{11}-\frac{2346304}{39214125}a^{10}+\frac{128387}{1568565}a^{9}+\frac{744932}{13071375}a^{8}-\frac{23032792}{39214125}a^{7}-\frac{679189}{484125}a^{6}+\frac{142969502}{39214125}a^{5}+\frac{18030238}{1452375}a^{4}+\frac{9803884}{871425}a^{3}-\frac{17600416}{1452375}a^{2}-\frac{8662927}{484125}a+\frac{6041636}{161375}$, $\frac{13472}{4705695}a^{11}-\frac{122164}{7842825}a^{10}+\frac{131066}{7842825}a^{9}+\frac{92942}{2614275}a^{8}-\frac{915811}{7842825}a^{7}-\frac{483424}{871425}a^{6}+\frac{14295902}{7842825}a^{5}+\frac{3599666}{871425}a^{4}-\frac{1027696}{871425}a^{3}-\frac{333692}{19365}a^{2}-\frac{521866}{32275}a-\frac{55132}{6455}$, $\frac{491218}{23528475}a^{11}-\frac{66128}{522855}a^{10}+\frac{2276393}{7842825}a^{9}-\frac{771593}{2614275}a^{8}-\frac{3744937}{7842825}a^{7}-\frac{238408}{104571}a^{6}+\frac{69746024}{7842825}a^{5}+\frac{28861136}{2614275}a^{4}+\frac{11800999}{871425}a^{3}-\frac{45597}{1291}a^{2}+\frac{1113908}{96825}a+\frac{977012}{32275}$, $\frac{18463073}{117642375}a^{11}-\frac{15312157}{39214125}a^{10}-\frac{5922313}{7842825}a^{9}+\frac{407911}{161375}a^{8}-\frac{166314251}{39214125}a^{7}-\frac{476061079}{13071375}a^{6}-\frac{200529014}{39214125}a^{5}+\frac{3577872536}{13071375}a^{4}+\frac{183929894}{290475}a^{3}+\frac{484069042}{1452375}a^{2}-\frac{405638396}{484125}a-\frac{182162222}{161375}$ Copy content Toggle raw display (assuming GRH)
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:  \( 307016.53712191305 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 307016.53712191305 \cdot 12}{2\cdot\sqrt{1772506678315561228125}}\cr\approx \mathstrut & 2.69214402006952 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 3*x^10 + 18*x^9 - 39*x^8 - 216*x^7 + 78*x^6 + 1647*x^5 + 2997*x^4 + 729*x^3 - 3645*x^2 + 6561)
 
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 - 3*x^10 + 18*x^9 - 39*x^8 - 216*x^7 + 78*x^6 + 1647*x^5 + 2997*x^4 + 729*x^3 - 3645*x^2 + 6561, 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 - 3*x^10 + 18*x^9 - 39*x^8 - 216*x^7 + 78*x^6 + 1647*x^5 + 2997*x^4 + 729*x^3 - 3645*x^2 + 6561);
 
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 - 3*x^10 + 18*x^9 - 39*x^8 - 216*x^7 + 78*x^6 + 1647*x^5 + 2997*x^4 + 729*x^3 - 3645*x^2 + 6561);
 
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

\(\Q(\sqrt{-51}) \), 3.1.135.1, 4.0.221085.2, 6.0.268618275.1

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: 12.2.2954177797192602046875.1
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 ${\href{/padicField/2.12.0.1}{12} }$ R R ${\href{/padicField/7.2.0.1}{2} }^{6}$ ${\href{/padicField/11.2.0.1}{2} }^{5}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ R ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.3.0.1}{3} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.12.0.1}{12} }$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{3}$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.2.0.1}{2} }^{6}$

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
\(3\) Copy content Toggle raw display 3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
\(17\) Copy content Toggle raw display 17.12.9.3$x^{12} + 289 x^{4} - 68782$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.15.2t1.a.a$1$ $ 3 \cdot 5 $ \(\Q(\sqrt{-15}) \) $C_2$ (as 2T1) $1$ $-1$
1.85.2t1.a.a$1$ $ 5 \cdot 17 $ \(\Q(\sqrt{85}) \) $C_2$ (as 2T1) $1$ $1$
* 1.51.2t1.a.a$1$ $ 3 \cdot 17 $ \(\Q(\sqrt{-51}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.135.3t2.b.a$2$ $ 3^{3} \cdot 5 $ 3.1.135.1 $S_3$ (as 3T2) $1$ $0$
* 2.39015.6t3.d.a$2$ $ 3^{3} \cdot 5 \cdot 17^{2}$ 6.2.447697125.2 $D_{6}$ (as 6T3) $1$ $0$
* 2.4335.4t3.d.a$2$ $ 3 \cdot 5 \cdot 17^{2}$ 4.2.368475.2 $D_{4}$ (as 4T3) $1$ $0$
* 2.39015.12t12.a.b$2$ $ 3^{3} \cdot 5 \cdot 17^{2}$ 12.0.1772506678315561228125.1 $D_{12}$ (as 12T12) $1$ $0$
* 2.39015.12t12.a.a$2$ $ 3^{3} \cdot 5 \cdot 17^{2}$ 12.0.1772506678315561228125.1 $D_{12}$ (as 12T12) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.