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Properties

Label 12.0.29365647704064.2
Degree $12$
Signature $[0, 6]$
Discriminant $2.937\times 10^{13}$
Root discriminant \(13.25\)
Ramified primes see page
Class number $2$
Class group $[2]$
Galois group $C_6\times S_3$ (as 12T18)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 10*x^10 - 16*x^9 + 20*x^8 - 20*x^7 + 18*x^6 - 20*x^5 + 7*x^4 + 4*x^3 + 4*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^12 - 4*x^11 + 10*x^10 - 16*x^9 + 20*x^8 - 20*x^7 + 18*x^6 - 20*x^5 + 7*x^4 + 4*x^3 + 4*x^2 - 4*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 4, 4, 7, -20, 18, -20, 20, -16, 10, -4, 1]);
 

\( x^{12} - 4 x^{11} + 10 x^{10} - 16 x^{9} + 20 x^{8} - 20 x^{7} + 18 x^{6} - 20 x^{5} + 7 x^{4} + 4 x^{3} + 4 x^{2} - 4 x + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(29365647704064\) \(\medspace = 2^{24}\cdot 3^{6}\cdot 7^{4}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(13.25\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:   \(2\), \(3\), \(7\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Aut(K/\Q) }$:  $6$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{15301}a^{11}-\frac{6605}{15301}a^{10}+\frac{7066}{15301}a^{9}-\frac{5234}{15301}a^{8}-\frac{4}{15301}a^{7}-\frac{4218}{15301}a^{6}-\frac{368}{1177}a^{5}-\frac{2100}{15301}a^{4}-\frac{599}{15301}a^{3}+\frac{6345}{15301}a^{2}-\frac{4504}{15301}a+\frac{1057}{15301}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

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);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
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);
 
Fundamental units:   $\frac{1181}{15301}a^{11}+\frac{3005}{15301}a^{10}-\frac{9400}{15301}a^{9}+\frac{30852}{15301}a^{8}-\frac{35326}{15301}a^{7}+\frac{52571}{15301}a^{6}-\frac{2649}{1177}a^{5}+\frac{44565}{15301}a^{4}-\frac{64777}{15301}a^{3}-\frac{34647}{15301}a^{2}+\frac{5524}{15301}a+\frac{8936}{15301}$, $\frac{12711}{15301}a^{11}-\frac{45471}{15301}a^{10}+\frac{106163}{15301}a^{9}-\frac{153636}{15301}a^{8}+\frac{178671}{15301}a^{7}-\frac{168605}{15301}a^{6}+\frac{11520}{1177}a^{5}-\frac{191768}{15301}a^{4}+\frac{6009}{15301}a^{3}+\frac{60928}{15301}a^{2}+\frac{82503}{15301}a-\frac{14052}{15301}$, $\frac{2594}{15301}a^{11}-\frac{11551}{15301}a^{10}+\frac{29208}{15301}a^{9}-\frac{50912}{15301}a^{8}+\frac{66129}{15301}a^{7}-\frac{77782}{15301}a^{6}+\frac{5840}{1177}a^{5}-\frac{92050}{15301}a^{4}+\frac{52799}{15301}a^{3}-\frac{4946}{15301}a^{2}+\frac{21889}{15301}a-\frac{12322}{15301}$, $\frac{10117}{15301}a^{11}-\frac{33920}{15301}a^{10}+\frac{76955}{15301}a^{9}-\frac{102724}{15301}a^{8}+\frac{112542}{15301}a^{7}-\frac{90823}{15301}a^{6}+\frac{5680}{1177}a^{5}-\frac{99718}{15301}a^{4}-\frac{46790}{15301}a^{3}+\frac{65874}{15301}a^{2}+\frac{60614}{15301}a-\frac{17031}{15301}$, $\frac{8936}{15301}a^{11}-\frac{36925}{15301}a^{10}+\frac{86355}{15301}a^{9}-\frac{133576}{15301}a^{8}+\frac{147868}{15301}a^{7}-\frac{143394}{15301}a^{6}+\frac{8329}{1177}a^{5}-\frac{144283}{15301}a^{4}+\frac{17987}{15301}a^{3}+\frac{100521}{15301}a^{2}+\frac{70391}{15301}a-\frac{25967}{15301}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 29.848139323999433 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 29.848139323999433 \cdot 2}{2\sqrt{29365647704064}}\approx 0.338903989446607$

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}, \sqrt{3})\), 6.0.677376.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: Deg 36
Degree 18 siblings: 18.0.86675003008018355847168.2, 18.6.292528135152061950984192.1

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 ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ R ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{6}$ ${\href{/padicField/13.2.0.1}{2} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.12.24.307$x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
\(3\) Copy content Toggle raw display 3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(7\) Copy content Toggle raw display 7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.8.2t1.b.a$1$ $ 2^{3}$ \(\Q(\sqrt{-2}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.24.2t1.b.a$1$ $ 2^{3} \cdot 3 $ \(\Q(\sqrt{-6}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.12.2t1.a.a$1$ $ 2^{2} \cdot 3 $ \(\Q(\sqrt{3}) \) $C_2$ (as 2T1) $1$ $1$
1.84.6t1.a.a$1$ $ 2^{2} \cdot 3 \cdot 7 $ 6.6.4148928.1 $C_6$ (as 6T1) $0$ $1$
1.56.6t1.c.a$1$ $ 2^{3} \cdot 7 $ 6.0.1229312.1 $C_6$ (as 6T1) $0$ $-1$
1.56.6t1.c.b$1$ $ 2^{3} \cdot 7 $ 6.0.1229312.1 $C_6$ (as 6T1) $0$ $-1$
1.168.6t1.c.a$1$ $ 2^{3} \cdot 3 \cdot 7 $ 6.0.33191424.1 $C_6$ (as 6T1) $0$ $-1$
1.84.6t1.a.b$1$ $ 2^{2} \cdot 3 \cdot 7 $ 6.6.4148928.1 $C_6$ (as 6T1) $0$ $1$
1.168.6t1.c.b$1$ $ 2^{3} \cdot 3 \cdot 7 $ 6.0.33191424.1 $C_6$ (as 6T1) $0$ $-1$
1.7.3t1.a.a$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
1.7.3t1.a.b$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
2.1176.3t2.a.a$2$ $ 2^{3} \cdot 3 \cdot 7^{2}$ 3.1.1176.1 $S_3$ (as 3T2) $1$ $0$
2.4704.6t3.f.a$2$ $ 2^{5} \cdot 3 \cdot 7^{2}$ 6.0.44255232.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.168.6t5.c.a$2$ $ 2^{3} \cdot 3 \cdot 7 $ 6.0.677376.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.168.6t5.c.b$2$ $ 2^{3} \cdot 3 \cdot 7 $ 6.0.677376.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.672.12t18.c.a$2$ $ 2^{5} \cdot 3 \cdot 7 $ 12.0.29365647704064.2 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.672.12t18.c.b$2$ $ 2^{5} \cdot 3 \cdot 7 $ 12.0.29365647704064.2 $C_6\times S_3$ (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 *.