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Properties

Label 16.0.53814899311...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{8}\cdot 23^{8}$
Root discriminant $72.14$
Ramified primes $2, 5, 23$
Class number $19200$ (GRH)
Class group $[2, 40, 240]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23710844, -14576384, 18896152, -9418640, 6750322, -2717984, 1377556, -452912, 177651, -48632, 15772, -3776, 1054, -224, 52, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 52*x^14 - 224*x^13 + 1054*x^12 - 3776*x^11 + 15772*x^10 - 48632*x^9 + 177651*x^8 - 452912*x^7 + 1377556*x^6 - 2717984*x^5 + 6750322*x^4 - 9418640*x^3 + 18896152*x^2 - 14576384*x + 23710844)
 
gp: K = bnfinit(x^16 - 8*x^15 + 52*x^14 - 224*x^13 + 1054*x^12 - 3776*x^11 + 15772*x^10 - 48632*x^9 + 177651*x^8 - 452912*x^7 + 1377556*x^6 - 2717984*x^5 + 6750322*x^4 - 9418640*x^3 + 18896152*x^2 - 14576384*x + 23710844, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 1054 x^{12} - 3776 x^{11} + 15772 x^{10} - 48632 x^{9} + 177651 x^{8} - 452912 x^{7} + 1377556 x^{6} - 2717984 x^{5} + 6750322 x^{4} - 9418640 x^{3} + 18896152 x^{2} - 14576384 x + 23710844 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(538148993119091792281600000000=2^{44}\cdot 5^{8}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.14$
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:  $2, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1840=2^{4}\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{1840}(1,·)$, $\chi_{1840}(321,·)$, $\chi_{1840}(1289,·)$, $\chi_{1840}(781,·)$, $\chi_{1840}(461,·)$, $\chi_{1840}(1749,·)$, $\chi_{1840}(1241,·)$, $\chi_{1840}(921,·)$, $\chi_{1840}(1701,·)$, $\chi_{1840}(229,·)$, $\chi_{1840}(1381,·)$, $\chi_{1840}(689,·)$, $\chi_{1840}(829,·)$, $\chi_{1840}(369,·)$, $\chi_{1840}(1609,·)$, $\chi_{1840}(1149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{62} a^{12} - \frac{3}{31} a^{11} - \frac{7}{62} a^{10} - \frac{3}{62} a^{9} + \frac{1}{31} a^{8} + \frac{3}{31} a^{7} - \frac{5}{62} a^{6} - \frac{27}{62} a^{5} - \frac{15}{62} a^{4} + \frac{12}{31} a^{3} + \frac{12}{31} a^{2} + \frac{3}{31} a + \frac{11}{31}$, $\frac{1}{62} a^{13} - \frac{6}{31} a^{11} - \frac{7}{31} a^{10} + \frac{15}{62} a^{9} - \frac{13}{62} a^{8} - \frac{13}{31} a^{6} - \frac{11}{31} a^{5} + \frac{27}{62} a^{4} - \frac{9}{31} a^{3} + \frac{13}{31} a^{2} - \frac{2}{31} a + \frac{4}{31}$, $\frac{1}{18089088480312094} a^{14} - \frac{1}{2584155497187442} a^{13} - \frac{10445320051753}{9044544240156047} a^{12} + \frac{125343840621127}{18089088480312094} a^{11} + \frac{1626549720373265}{9044544240156047} a^{10} + \frac{674606070885613}{18089088480312094} a^{9} + \frac{309173366894705}{9044544240156047} a^{8} - \frac{5142241113638635}{18089088480312094} a^{7} + \frac{1113396484475901}{18089088480312094} a^{6} - \frac{2797344240590058}{9044544240156047} a^{5} + \frac{3723661526901627}{9044544240156047} a^{4} + \frac{2929910428536248}{9044544240156047} a^{3} - \frac{3661592172752242}{9044544240156047} a^{2} - \frac{505465950483792}{9044544240156047} a + \frac{3340251286560417}{9044544240156047}$, $\frac{1}{20877141598694116736126} a^{15} + \frac{577057}{20877141598694116736126} a^{14} + \frac{13660460711721356800}{10438570799347058368063} a^{13} + \frac{147086854263104479015}{20877141598694116736126} a^{12} - \frac{692576985550670498541}{2982448799813445248018} a^{11} + \frac{1765751360559231848983}{20877141598694116736126} a^{10} + \frac{4919754180322483015043}{20877141598694116736126} a^{9} + \frac{3298770912366990598491}{20877141598694116736126} a^{8} - \frac{2467506705689770938350}{10438570799347058368063} a^{7} - \frac{2069918254351275745201}{10438570799347058368063} a^{6} + \frac{911390217210847625095}{20877141598694116736126} a^{5} + \frac{3401901253528920052313}{10438570799347058368063} a^{4} - \frac{669713739831610171892}{1491224399906722624009} a^{3} + \frac{3937170690284188059344}{10438570799347058368063} a^{2} - \frac{705781485844404639426}{10438570799347058368063} a - \frac{5066464913853061116156}{10438570799347058368063}$

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

Class group and class number

$C_{2}\times C_{40}\times C_{240}$, which has order $19200$ (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:  $7$
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:  \( 12198.951274811623 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-46}) \), \(\Q(\sqrt{-115}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-230}) \), \(\Q(\sqrt{10}, \sqrt{-46})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{10}, \sqrt{-23})\), \(\Q(\sqrt{2}, \sqrt{-23})\), \(\Q(\sqrt{5}, \sqrt{-46})\), \(\Q(\sqrt{2}, \sqrt{-115})\), \(\Q(\sqrt{5}, \sqrt{-23})\), 4.0.27084800.5, 4.0.1083392.5, 4.4.51200.1, \(\Q(\zeta_{16})^+\), 8.0.716392960000.2, 8.0.733586391040000.76, 8.8.2621440000.1, 8.0.733586391040000.91, 8.0.1173738225664.2, 8.0.733586391040000.71, 8.0.733586391040000.42

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$