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Properties

Label 12.0.941480149401.1
Degree $12$
Signature $[0, 6]$
Discriminant $941480149401$
Root discriminant \(9.95\)
Ramified primes $3,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_6\times S_3$ (as 12T18)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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

\( x^{12} + x^{10} - 4x^{9} - 2x^{8} - 3x^{7} + 9x^{6} + 4x^{5} + 12x^{4} - 7x^{3} - 2x^{2} - x + 1 \) 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:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(941480149401\) \(\medspace = 3^{12}\cdot 11^{6}\) 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:  \(9.95\)
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:  $3^{3/2}11^{1/2}\approx 17.233687939614086$
Ramified primes:   \(3\), \(11\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6}a^{9}-\frac{1}{3}a^{7}-\frac{1}{2}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{6}$, $\frac{1}{30}a^{10}+\frac{1}{15}a^{9}-\frac{7}{15}a^{8}+\frac{1}{6}a^{7}+\frac{2}{15}a^{6}+\frac{4}{15}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{13}{30}a+\frac{2}{15}$, $\frac{1}{90}a^{11}+\frac{1}{90}a^{10}-\frac{1}{90}a^{9}+\frac{19}{90}a^{8}-\frac{31}{90}a^{7}-\frac{19}{90}a^{6}-\frac{11}{45}a^{5}+\frac{2}{15}a^{4}+\frac{1}{3}a^{3}-\frac{37}{90}a^{2}-\frac{13}{30}a+\frac{41}{90}$ 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 $1$ (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:   \( \frac{1}{2} a^{11} + \frac{1}{2} a^{10} + \frac{2}{3} a^{9} - \frac{3}{2} a^{8} - \frac{17}{6} a^{7} - 3 a^{6} + \frac{8}{3} a^{5} + \frac{17}{3} a^{4} + 9 a^{3} + \frac{17}{6} a^{2} - \frac{11}{6} a - \frac{2}{3} \)  (order $6$) 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:   $\frac{11}{15}a^{11}+\frac{3}{5}a^{10}+a^{9}-\frac{11}{5}a^{8}-\frac{17}{5}a^{7}-\frac{67}{15}a^{6}+\frac{58}{15}a^{5}+\frac{101}{15}a^{4}+\frac{40}{3}a^{3}+\frac{21}{5}a^{2}-\frac{4}{3}a-\frac{37}{15}$, $\frac{7}{15}a^{11}+\frac{1}{15}a^{10}+\frac{17}{30}a^{9}-\frac{23}{15}a^{8}-\frac{17}{15}a^{7}-\frac{59}{30}a^{6}+\frac{46}{15}a^{5}+\frac{26}{15}a^{4}+7a^{3}-\frac{3}{5}a^{2}-\frac{1}{15}a-\frac{3}{10}$, $\frac{28}{45}a^{11}+\frac{7}{18}a^{10}+\frac{41}{45}a^{9}-\frac{86}{45}a^{8}-\frac{221}{90}a^{7}-\frac{169}{45}a^{6}+\frac{149}{45}a^{5}+\frac{23}{5}a^{4}+11a^{3}+\frac{104}{45}a^{2}-\frac{3}{10}a-\frac{64}{45}$, $\frac{11}{45}a^{11}+\frac{13}{90}a^{10}+\frac{7}{18}a^{9}-\frac{43}{45}a^{8}-\frac{67}{90}a^{7}-\frac{139}{90}a^{6}+\frac{88}{45}a^{5}+\frac{22}{15}a^{4}+\frac{13}{3}a^{3}-\frac{17}{45}a^{2}+\frac{3}{2}a-\frac{109}{90}$, $\frac{1}{45}a^{11}+\frac{7}{18}a^{10}+\frac{2}{45}a^{9}+\frac{13}{45}a^{8}-\frac{137}{90}a^{7}-\frac{43}{45}a^{6}-\frac{52}{45}a^{5}+\frac{53}{15}a^{4}+2a^{3}+\frac{233}{45}a^{2}-\frac{53}{30}a-\frac{43}{45}$ 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:  \( 27.6923649209 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 27.6923649209 \cdot 1}{6\cdot\sqrt{941480149401}}\cr\approx \mathstrut & 0.292672840071 \end{aligned}\] (assuming GRH)

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

$C_6\times S_3$ (as 12T18):

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 36
The 18 conjugacy class representatives for $C_6\times S_3$
Character table for $C_6\times S_3$

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-3}, \sqrt{-11})\), 6.0.107811.1

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 36
Degree 18 siblings: 18.6.17980759982220503815017.1, 18.0.13509211106101054707.1
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 ${\href{/padicField/2.6.0.1}{6} }^{2}$ R ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ ${\href{/padicField/7.6.0.1}{6} }^{2}$ R ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{6}$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$

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 3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
3.6.3.2$x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(11\) Copy content Toggle raw display 11.12.6.1$x^{12} + 72 x^{10} + 8 x^{9} + 2034 x^{8} + 38 x^{7} + 27996 x^{6} - 6312 x^{5} + 196025 x^{4} - 84710 x^{3} + 695581 x^{2} - 235284 x + 1080083$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.33.2t1.a.a$1$ $ 3 \cdot 11 $ \(\Q(\sqrt{33}) \) $C_2$ (as 2T1) $1$ $1$
* 1.11.2t1.a.a$1$ $ 11 $ \(\Q(\sqrt{-11}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.99.6t1.a.a$1$ $ 3^{2} \cdot 11 $ 6.0.8732691.1 $C_6$ (as 6T1) $0$ $-1$
1.9.6t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})\) $C_6$ (as 6T1) $0$ $-1$
1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.99.6t1.b.a$1$ $ 3^{2} \cdot 11 $ 6.6.26198073.1 $C_6$ (as 6T1) $0$ $1$
1.99.6t1.a.b$1$ $ 3^{2} \cdot 11 $ 6.0.8732691.1 $C_6$ (as 6T1) $0$ $-1$
1.99.6t1.b.b$1$ $ 3^{2} \cdot 11 $ 6.6.26198073.1 $C_6$ (as 6T1) $0$ $1$
1.9.6t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})\) $C_6$ (as 6T1) $0$ $-1$
2.891.3t2.b.a$2$ $ 3^{4} \cdot 11 $ 3.1.891.1 $S_3$ (as 3T2) $1$ $0$
2.891.6t3.b.a$2$ $ 3^{4} \cdot 11 $ 6.2.26198073.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.99.6t5.a.a$2$ $ 3^{2} \cdot 11 $ 6.0.107811.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.99.6t5.a.b$2$ $ 3^{2} \cdot 11 $ 6.0.107811.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.297.12t18.b.a$2$ $ 3^{3} \cdot 11 $ 12.0.941480149401.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.297.12t18.b.b$2$ $ 3^{3} \cdot 11 $ 12.0.941480149401.1 $C_6\times S_3$ (as 12T18) $0$ $0$

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 *.