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Properties

Label 12.0.71228826095703125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 19^{4}\cdot 23^{4}$
Root discriminant $25.37$
Ramified primes $5, 19, 23$
Class number $3$
Class group $[3]$
Galois group $C_2\times C_3:S_3.C_2$ (as 12T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -150, 360, 30, 1546, 246, 267, 43, 50, -8, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 7*x^10 - 8*x^9 + 50*x^8 + 43*x^7 + 267*x^6 + 246*x^5 + 1546*x^4 + 30*x^3 + 360*x^2 - 150*x + 25)
 
gp: K = bnfinit(x^12 - x^11 + 7*x^10 - 8*x^9 + 50*x^8 + 43*x^7 + 267*x^6 + 246*x^5 + 1546*x^4 + 30*x^3 + 360*x^2 - 150*x + 25, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 7 x^{10} - 8 x^{9} + 50 x^{8} + 43 x^{7} + 267 x^{6} + 246 x^{5} + 1546 x^{4} + 30 x^{3} + 360 x^{2} - 150 x + 25 \)

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:  \(71228826095703125=5^{9}\cdot 19^{4}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.37$
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:  $5, 19, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{2}$, $\frac{1}{15} a^{8} + \frac{1}{15} a^{7} - \frac{1}{15} a^{6} + \frac{1}{5} a^{5} + \frac{1}{15} a^{4} - \frac{1}{5} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{30} a^{9} - \frac{1}{15} a^{7} + \frac{1}{30} a^{6} + \frac{7}{30} a^{5} - \frac{13}{30} a^{4} - \frac{2}{5} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{60} a^{10} - \frac{1}{60} a^{9} - \frac{1}{60} a^{7} - \frac{1}{30} a^{6} - \frac{13}{30} a^{5} + \frac{3}{20} a^{4} - \frac{4}{15} a^{3} + \frac{1}{4} a^{2} + \frac{1}{6} a + \frac{1}{4}$, $\frac{1}{538449217080} a^{11} + \frac{723497481}{89741536180} a^{10} - \frac{2815491431}{538449217080} a^{9} - \frac{2728498811}{179483072360} a^{8} + \frac{11853629169}{179483072360} a^{7} + \frac{256896956}{4487076809} a^{6} - \frac{1088900271}{179483072360} a^{5} + \frac{46391157743}{107689843416} a^{4} + \frac{41265817127}{538449217080} a^{3} - \frac{79135138897}{538449217080} a^{2} - \frac{39166916275}{107689843416} a + \frac{318756373}{107689843416}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( \frac{435704577}{179483072360} a^{11} - \frac{17592691}{13461230427} a^{10} + \frac{9476239213}{538449217080} a^{9} - \frac{7118260819}{538449217080} a^{8} + \frac{22311065777}{179483072360} a^{7} + \frac{6460524079}{44870768090} a^{6} + \frac{84112466761}{107689843416} a^{5} + \frac{527056857959}{538449217080} a^{4} + \frac{2409401203333}{538449217080} a^{3} + \frac{404387054141}{179483072360} a^{2} + \frac{117656979233}{35896614472} a + \frac{21377984857}{107689843416} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5327.46417737 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3.C_2$ (as 12T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 12 conjugacy class representatives for $C_2\times C_3:S_3.C_2$
Character table for $C_2\times C_3:S_3.C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 6.6.119355625.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ R R ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
19.6.4.2$x^{6} - 19 x^{3} + 722$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$23$23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$