↑

Properties

Label 12.0.17523242277...8125.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{16}\cdot 5^{9}\cdot 7^{6}\cdot 11^{6}$
Root discriminant $126.95$
Ramified primes $3, 5, 7, 11$
Class number $114256$ (GRH)
Class group $[2, 2, 28564]$ (GRH)
Galois group $C_{12}$ (as 12T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9174406301, -1176569685, 1226604198, -119458244, 68420724, -4813584, 2005158, -91308, 32505, -831, 279, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 279*x^10 - 831*x^9 + 32505*x^8 - 91308*x^7 + 2005158*x^6 - 4813584*x^5 + 68420724*x^4 - 119458244*x^3 + 1226604198*x^2 - 1176569685*x + 9174406301)
 
gp: K = bnfinit(x^12 - 3*x^11 + 279*x^10 - 831*x^9 + 32505*x^8 - 91308*x^7 + 2005158*x^6 - 4813584*x^5 + 68420724*x^4 - 119458244*x^3 + 1226604198*x^2 - 1176569685*x + 9174406301, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + 279 x^{10} - 831 x^{9} + 32505 x^{8} - 91308 x^{7} + 2005158 x^{6} - 4813584 x^{5} + 68420724 x^{4} - 119458244 x^{3} + 1226604198 x^{2} - 1176569685 x + 9174406301 \)

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:  \(17523242277045191736328125=3^{16}\cdot 5^{9}\cdot 7^{6}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $126.95$
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:  $3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3465=3^{2}\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3465}(2848,·)$, $\chi_{3465}(1,·)$, $\chi_{3465}(1156,·)$, $\chi_{3465}(1462,·)$, $\chi_{3465}(2311,·)$, $\chi_{3465}(307,·)$, $\chi_{3465}(694,·)$, $\chi_{3465}(1849,·)$, $\chi_{3465}(2617,·)$, $\chi_{3465}(538,·)$, $\chi_{3465}(3004,·)$, $\chi_{3465}(1693,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{419} a^{9} - \frac{60}{419} a^{8} + \frac{105}{419} a^{7} + \frac{63}{419} a^{6} - \frac{193}{419} a^{5} + \frac{59}{419} a^{4} + \frac{58}{419} a^{3} + \frac{154}{419} a^{2} - \frac{44}{419} a + \frac{54}{419}$, $\frac{1}{419} a^{10} - \frac{143}{419} a^{8} + \frac{78}{419} a^{7} - \frac{184}{419} a^{6} - \frac{208}{419} a^{5} - \frac{173}{419} a^{4} - \frac{137}{419} a^{3} - \frac{22}{419} a^{2} - \frac{72}{419} a - \frac{112}{419}$, $\frac{1}{11448038200989648908463798165533587849} a^{11} + \frac{346791403535133898649528168796193}{11448038200989648908463798165533587849} a^{10} - \frac{5052015180423586879783762113868216}{11448038200989648908463798165533587849} a^{9} + \frac{1031724301225725317116351670968086031}{11448038200989648908463798165533587849} a^{8} - \frac{1040118372758143104500885953106552654}{11448038200989648908463798165533587849} a^{7} - \frac{1234625590892691249935892891602468580}{11448038200989648908463798165533587849} a^{6} + \frac{1002617680865813447282794304251611248}{11448038200989648908463798165533587849} a^{5} + \frac{3933437913563356449948326181877598082}{11448038200989648908463798165533587849} a^{4} + \frac{4751507711870855993235726339586363025}{11448038200989648908463798165533587849} a^{3} - \frac{4818900035928199096967354309065585399}{11448038200989648908463798165533587849} a^{2} - \frac{4087026224112998914948636453835438264}{11448038200989648908463798165533587849} a + \frac{3789798229428210824088799685371928819}{11448038200989648908463798165533587849}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{28564}$, which has order $114256$ (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:  \( -1 \) (order $2$)
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:  \( 201.000834787 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.0.741125.2, 6.6.820125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ R R R R ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$3$3.12.16.14$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$$3$$4$$16$$C_{12}$$[2]^{4}$
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.6.2$x^{12} + 7203 x^{4} - 16807 x^{2} + 588245$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$