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Properties

Label 12.0.13217819187...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 127^{8}$
Root discriminant $84.48$
Ramified primes $5, 127$
Class number $148$ (GRH)
Class group $[2, 74]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![40960000, 21504000, 10777600, 4877440, 2157136, 330408, 85532, 13810, 2051, -165, 43, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 43*x^10 - 165*x^9 + 2051*x^8 + 13810*x^7 + 85532*x^6 + 330408*x^5 + 2157136*x^4 + 4877440*x^3 + 10777600*x^2 + 21504000*x + 40960000)
 
gp: K = bnfinit(x^12 - x^11 + 43*x^10 - 165*x^9 + 2051*x^8 + 13810*x^7 + 85532*x^6 + 330408*x^5 + 2157136*x^4 + 4877440*x^3 + 10777600*x^2 + 21504000*x + 40960000, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 43 x^{10} - 165 x^{9} + 2051 x^{8} + 13810 x^{7} + 85532 x^{6} + 330408 x^{5} + 2157136 x^{4} + 4877440 x^{3} + 10777600 x^{2} + 21504000 x + 40960000 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(132178191876990001953125=5^{9}\cdot 127^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.48$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(635=5\cdot 127\)
Dirichlet character group:    $\lbrace$$\chi_{635}(128,·)$, $\chi_{635}(1,·)$, $\chi_{635}(488,·)$, $\chi_{635}(361,·)$, $\chi_{635}(234,·)$, $\chi_{635}(107,·)$, $\chi_{635}(527,·)$, $\chi_{635}(273,·)$, $\chi_{635}(146,·)$, $\chi_{635}(19,·)$, $\chi_{635}(509,·)$, $\chi_{635}(382,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} + \frac{7}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{209162541440} a^{9} - \frac{265414601}{209162541440} a^{8} + \frac{6397007923}{209162541440} a^{7} - \frac{3508884249}{41832508288} a^{6} - \frac{6291890069}{209162541440} a^{5} + \frac{468633705}{20916254144} a^{4} + \frac{11299919683}{52290635360} a^{3} - \frac{4022647679}{26145317680} a^{2} + \frac{5720183381}{13072658840} a + \frac{102149082}{326816471}$, $\frac{1}{8366501657600} a^{10} - \frac{1}{8366501657600} a^{9} + \frac{98963988843}{8366501657600} a^{8} + \frac{5126201247}{1673300331520} a^{7} - \frac{52394303549}{8366501657600} a^{6} + \frac{136116209861}{836650165760} a^{5} + \frac{688574577783}{2091625414400} a^{4} - \frac{163338907099}{1045812707200} a^{3} - \frac{152736275179}{522906353600} a^{2} - \frac{1360395939}{13072658840} a - \frac{1114926}{326816471}$, $\frac{1}{334660066304000} a^{11} - \frac{1}{334660066304000} a^{10} + \frac{43}{334660066304000} a^{9} + \frac{1253869222367}{66932013260800} a^{8} - \frac{5927251645949}{334660066304000} a^{7} + \frac{1384437076581}{33466006630400} a^{6} + \frac{18078384229383}{83665016576000} a^{5} - \frac{2581253998699}{41832508288000} a^{4} + \frac{7586370854821}{20916254144000} a^{3} + \frac{185704190021}{522906353600} a^{2} + \frac{246473081}{13072658840} a + \frac{124448422}{326816471}$

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

Class group and class number

$C_{2}\times C_{74}$, which has order $148$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{21043491}{334660066304000} a^{11} + \frac{61126331}{334660066304000} a^{10} - \frac{944952953}{334660066304000} a^{9} + \frac{1015006827}{66932013260800} a^{8} - \frac{49773868641}{334660066304000} a^{7} - \frac{20840070587}{33466006630400} a^{6} - \frac{311586962953}{83665016576000} a^{5} - \frac{440571537931}{41832508288000} a^{4} - \frac{2179122172691}{20916254144000} a^{3} - \frac{1263611531}{26145317680} a^{2} - \frac{61126331}{653632942} a - \frac{20041420}{326816471} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 162763.773713 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.16129.1, \(\Q(\zeta_{5})\), 6.6.32518080125.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$127$127.12.8.1$x^{12} - 381 x^{9} + 48387 x^{6} - 2048383 x^{3} + 7023905307$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.4t1.1c1$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.5.4t1.1c2$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
* 1.127.3t1.1c1$1$ $ 127 $ $x^{3} - x^{2} - 42 x - 80$ $C_3$ (as 3T1) $0$ $1$
* 1.5_127.12t1.1c1$1$ $ 5 \cdot 127 $ $x^{12} - x^{11} + 43 x^{10} - 165 x^{9} + 2051 x^{8} + 13810 x^{7} + 85532 x^{6} + 330408 x^{5} + 2157136 x^{4} + 4877440 x^{3} + 10777600 x^{2} + 21504000 x + 40960000$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.5_127.6t1.1c1$1$ $ 5 \cdot 127 $ $x^{6} - x^{5} - 88 x^{4} + 247 x^{3} + 1688 x^{2} - 7841 x + 7999$ $C_6$ (as 6T1) $0$ $1$
* 1.5_127.12t1.1c2$1$ $ 5 \cdot 127 $ $x^{12} - x^{11} + 43 x^{10} - 165 x^{9} + 2051 x^{8} + 13810 x^{7} + 85532 x^{6} + 330408 x^{5} + 2157136 x^{4} + 4877440 x^{3} + 10777600 x^{2} + 21504000 x + 40960000$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.127.3t1.1c2$1$ $ 127 $ $x^{3} - x^{2} - 42 x - 80$ $C_3$ (as 3T1) $0$ $1$
* 1.5_127.12t1.1c3$1$ $ 5 \cdot 127 $ $x^{12} - x^{11} + 43 x^{10} - 165 x^{9} + 2051 x^{8} + 13810 x^{7} + 85532 x^{6} + 330408 x^{5} + 2157136 x^{4} + 4877440 x^{3} + 10777600 x^{2} + 21504000 x + 40960000$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.5_127.6t1.1c2$1$ $ 5 \cdot 127 $ $x^{6} - x^{5} - 88 x^{4} + 247 x^{3} + 1688 x^{2} - 7841 x + 7999$ $C_6$ (as 6T1) $0$ $1$
* 1.5_127.12t1.1c4$1$ $ 5 \cdot 127 $ $x^{12} - x^{11} + 43 x^{10} - 165 x^{9} + 2051 x^{8} + 13810 x^{7} + 85532 x^{6} + 330408 x^{5} + 2157136 x^{4} + 4877440 x^{3} + 10777600 x^{2} + 21504000 x + 40960000$ $C_{12}$ (as 12T1) $0$ $-1$

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 *.