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Properties

Label 12.0.68271795275...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 5^{9}\cdot 7^{8}\cdot 23^{6}$
Root discriminant $117.36$
Ramified primes $2, 5, 7, 23$
Class number $125540$ (GRH)
Class group $[2, 62770]$ (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![22837898761, -1800348698, 2704663170, -173414484, 132905105, -6656386, 3418417, -124252, 47714, -1126, 343, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 343*x^10 - 1126*x^9 + 47714*x^8 - 124252*x^7 + 3418417*x^6 - 6656386*x^5 + 132905105*x^4 - 173414484*x^3 + 2704663170*x^2 - 1800348698*x + 22837898761)
 
gp: K = bnfinit(x^12 - 4*x^11 + 343*x^10 - 1126*x^9 + 47714*x^8 - 124252*x^7 + 3418417*x^6 - 6656386*x^5 + 132905105*x^4 - 173414484*x^3 + 2704663170*x^2 - 1800348698*x + 22837898761, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 343 x^{10} - 1126 x^{9} + 47714 x^{8} - 124252 x^{7} + 3418417 x^{6} - 6656386 x^{5} + 132905105 x^{4} - 173414484 x^{3} + 2704663170 x^{2} - 1800348698 x + 22837898761 \)

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:  \(6827179527544712000000000=2^{12}\cdot 5^{9}\cdot 7^{8}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $117.36$
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:  $2, 5, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3220=2^{2}\cdot 5\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{3220}(1,·)$, $\chi_{3220}(1563,·)$, $\chi_{3220}(1381,·)$, $\chi_{3220}(2209,·)$, $\chi_{3220}(1289,·)$, $\chi_{3220}(2669,·)$, $\chi_{3220}(1103,·)$, $\chi_{3220}(1747,·)$, $\chi_{3220}(183,·)$, $\chi_{3220}(921,·)$, $\chi_{3220}(827,·)$, $\chi_{3220}(2207,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{23} a^{6} - \frac{2}{23} a^{5} - \frac{3}{23} a^{4} + \frac{6}{23} a^{3} + \frac{2}{23} a^{2} - \frac{4}{23} a + \frac{1}{23}$, $\frac{1}{23} a^{7} - \frac{7}{23} a^{5} - \frac{9}{23} a^{3} - \frac{7}{23} a + \frac{2}{23}$, $\frac{1}{23} a^{8} + \frac{9}{23} a^{5} - \frac{7}{23} a^{4} - \frac{4}{23} a^{3} + \frac{7}{23} a^{2} - \frac{3}{23} a + \frac{7}{23}$, $\frac{1}{23} a^{9} + \frac{11}{23} a^{5} - \frac{1}{23} a^{3} + \frac{2}{23} a^{2} - \frac{3}{23} a - \frac{9}{23}$, $\frac{1}{1139352172103} a^{10} - \frac{20806941616}{1139352172103} a^{9} + \frac{6087284113}{1139352172103} a^{8} + \frac{16492897723}{1139352172103} a^{7} - \frac{16686069814}{1139352172103} a^{6} - \frac{372442820778}{1139352172103} a^{5} + \frac{422114670793}{1139352172103} a^{4} + \frac{356886078144}{1139352172103} a^{3} - \frac{20470744049}{49537050961} a^{2} + \frac{276992754248}{1139352172103} a - \frac{400022280621}{1139352172103}$, $\frac{1}{100215207721858228106477323} a^{11} - \frac{5248711016065}{100215207721858228106477323} a^{10} + \frac{1319470891699998328138424}{100215207721858228106477323} a^{9} + \frac{1309808562160901024226726}{100215207721858228106477323} a^{8} - \frac{9558535196480784387520}{100215207721858228106477323} a^{7} + \frac{48021287187918674349474}{4357182944428618613325101} a^{6} - \frac{41049522588761074489347343}{100215207721858228106477323} a^{5} + \frac{13876223950401217769562312}{100215207721858228106477323} a^{4} + \frac{43524241637457695371156542}{100215207721858228106477323} a^{3} + \frac{8502992931773043170261178}{100215207721858228106477323} a^{2} + \frac{12899591623111750177843996}{100215207721858228106477323} a - \frac{5302526072726728671650252}{100215207721858228106477323}$

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

Class group and class number

$C_{2}\times C_{62770}$, which has order $125540$ (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:  \( 104.882003477 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.1058000.1, 6.6.300125.1

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 R ${\href{/LocalNumberField/3.12.0.1}{12} }$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$2$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$23$23.12.6.2$x^{12} - 6436343 x^{2} + 2220538335$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$