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Properties

Label 12.0.19122433828...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 7^{8}\cdot 19^{8}$
Root discriminant $87.12$
Ramified primes $5, 7, 19$
Class number $444$ (GRH)
Class group $[2, 222]$ (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![16777216, -11534336, 8192000, -5550080, 3763456, -169280, 88464, -4660, 1941, -25, 45, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 45*x^10 - 25*x^9 + 1941*x^8 - 4660*x^7 + 88464*x^6 - 169280*x^5 + 3763456*x^4 - 5550080*x^3 + 8192000*x^2 - 11534336*x + 16777216)
 
gp: K = bnfinit(x^12 - x^11 + 45*x^10 - 25*x^9 + 1941*x^8 - 4660*x^7 + 88464*x^6 - 169280*x^5 + 3763456*x^4 - 5550080*x^3 + 8192000*x^2 - 11534336*x + 16777216, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 45 x^{10} - 25 x^{9} + 1941 x^{8} - 4660 x^{7} + 88464 x^{6} - 169280 x^{5} + 3763456 x^{4} - 5550080 x^{3} + 8192000 x^{2} - 11534336 x + 16777216 \)

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:  \(191224338285780939453125=5^{9}\cdot 7^{8}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.12$
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:  $5, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(665=5\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{665}(1,·)$, $\chi_{665}(387,·)$, $\chi_{665}(134,·)$, $\chi_{665}(267,·)$, $\chi_{665}(653,·)$, $\chi_{665}(144,·)$, $\chi_{665}(11,·)$, $\chi_{665}(533,·)$, $\chi_{665}(121,·)$, $\chi_{665}(543,·)$, $\chi_{665}(254,·)$, $\chi_{665}(277,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{3}{16} a^{4} + \frac{7}{16} a^{3} + \frac{5}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{64} a^{7} - \frac{1}{64} a^{6} - \frac{3}{64} a^{5} + \frac{23}{64} a^{4} - \frac{27}{64} a^{3} - \frac{1}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{256} a^{8} - \frac{1}{256} a^{7} - \frac{3}{256} a^{6} + \frac{23}{256} a^{5} + \frac{37}{256} a^{4} + \frac{31}{64} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{114657857536} a^{9} + \frac{85097551}{114657857536} a^{8} - \frac{139554083}{114657857536} a^{7} - \frac{3282269705}{114657857536} a^{6} + \frac{2588133317}{114657857536} a^{5} + \frac{13972769047}{28664464384} a^{4} + \frac{2737694165}{7166116096} a^{3} - \frac{18408549}{1791529024} a^{2} + \frac{91659773}{447882256} a - \frac{5784456}{27992641}$, $\frac{1}{1834525720576} a^{10} - \frac{1}{1834525720576} a^{9} + \frac{3385989933}{1834525720576} a^{8} - \frac{8706120473}{1834525720576} a^{7} + \frac{43031819925}{1834525720576} a^{6} - \frac{28056271181}{458631430144} a^{5} - \frac{24130766839}{114657857536} a^{4} + \frac{12759042299}{28664464384} a^{3} + \frac{1901103565}{7166116096} a^{2} + \frac{17015285}{447882256} a - \frac{9792254}{27992641}$, $\frac{1}{29352411529216} a^{11} - \frac{1}{29352411529216} a^{10} + \frac{45}{29352411529216} a^{9} - \frac{17556987929}{29352411529216} a^{8} - \frac{122841290859}{29352411529216} a^{7} + \frac{91192341363}{7338102882304} a^{6} + \frac{27791843225}{1834525720576} a^{5} + \frac{120989687083}{458631430144} a^{4} + \frac{46495815725}{114657857536} a^{3} + \frac{1415118565}{7166116096} a^{2} + \frac{181677717}{447882256} a + \frac{13614133}{27992641}$

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

Class group and class number

$C_{2}\times C_{222}$, which has order $444$ (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:  \( \frac{20951}{7338102882304} a^{11} - \frac{942795}{7338102882304} a^{10} + \frac{523775}{7338102882304} a^{9} - \frac{40665891}{7338102882304} a^{8} + \frac{4208695}{7338102882304} a^{7} - \frac{115838079}{458631430144} a^{6} + \frac{55415395}{114657857536} a^{5} - \frac{308000651}{28664464384} a^{4} + \frac{28388605}{1791529024} a^{3} - \frac{3447376625}{7166116096} a^{2} + \frac{921844}{27992641} a - \frac{1340864}{27992641} \) (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:  \( 49470.5066478 \) (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

\(\Q(\sqrt{5}) \), 3.3.17689.2, \(\Q(\zeta_{5})\), 6.6.39112590125.1

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 R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ R ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$$[\ ]_{4}$
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$19$19.6.4.2$x^{6} - 19 x^{3} + 722$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.2$x^{6} - 19 x^{3} + 722$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

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) $0$ $-1$
* 1.7_19.3t1.2c1$1$ $ 7 \cdot 19 $ $x^{3} - x^{2} - 44 x + 64$ $C_3$ (as 3T1) $0$ $1$
* 1.5_7_19.12t1.1c1$1$ $ 5 \cdot 7 \cdot 19 $ $x^{12} - x^{11} + 45 x^{10} - 25 x^{9} + 1941 x^{8} - 4660 x^{7} + 88464 x^{6} - 169280 x^{5} + 3763456 x^{4} - 5550080 x^{3} + 8192000 x^{2} - 11534336 x + 16777216$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.5_7_19.6t1.3c1$1$ $ 5 \cdot 7 \cdot 19 $ $x^{6} - x^{5} - 92 x^{4} - 37 x^{3} + 2004 x^{2} + 3335 x - 505$ $C_6$ (as 6T1) $0$ $1$
* 1.5_7_19.12t1.1c2$1$ $ 5 \cdot 7 \cdot 19 $ $x^{12} - x^{11} + 45 x^{10} - 25 x^{9} + 1941 x^{8} - 4660 x^{7} + 88464 x^{6} - 169280 x^{5} + 3763456 x^{4} - 5550080 x^{3} + 8192000 x^{2} - 11534336 x + 16777216$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.7_19.3t1.2c2$1$ $ 7 \cdot 19 $ $x^{3} - x^{2} - 44 x + 64$ $C_3$ (as 3T1) $0$ $1$
* 1.5_7_19.12t1.1c3$1$ $ 5 \cdot 7 \cdot 19 $ $x^{12} - x^{11} + 45 x^{10} - 25 x^{9} + 1941 x^{8} - 4660 x^{7} + 88464 x^{6} - 169280 x^{5} + 3763456 x^{4} - 5550080 x^{3} + 8192000 x^{2} - 11534336 x + 16777216$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.5_7_19.6t1.3c2$1$ $ 5 \cdot 7 \cdot 19 $ $x^{6} - x^{5} - 92 x^{4} - 37 x^{3} + 2004 x^{2} + 3335 x - 505$ $C_6$ (as 6T1) $0$ $1$
* 1.5_7_19.12t1.1c4$1$ $ 5 \cdot 7 \cdot 19 $ $x^{12} - x^{11} + 45 x^{10} - 25 x^{9} + 1941 x^{8} - 4660 x^{7} + 88464 x^{6} - 169280 x^{5} + 3763456 x^{4} - 5550080 x^{3} + 8192000 x^{2} - 11534336 x + 16777216$ $C_{12}$ (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 *.