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Properties

Label 16.0.24759631762...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{12}\cdot 7^{8}$
Root discriminant $59.51$
Ramified primes $2, 5, 7$
Class number $340$ (GRH)
Class group $[2, 170]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1113751, 228044, -261314, -604188, 347147, 29700, -772, -26568, -3713, 7832, -100, -1140, 135, 84, -14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 14*x^14 + 84*x^13 + 135*x^12 - 1140*x^11 - 100*x^10 + 7832*x^9 - 3713*x^8 - 26568*x^7 - 772*x^6 + 29700*x^5 + 347147*x^4 - 604188*x^3 - 261314*x^2 + 228044*x + 1113751)
 
gp: K = bnfinit(x^16 - 4*x^15 - 14*x^14 + 84*x^13 + 135*x^12 - 1140*x^11 - 100*x^10 + 7832*x^9 - 3713*x^8 - 26568*x^7 - 772*x^6 + 29700*x^5 + 347147*x^4 - 604188*x^3 - 261314*x^2 + 228044*x + 1113751, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 14 x^{14} + 84 x^{13} + 135 x^{12} - 1140 x^{11} - 100 x^{10} + 7832 x^{9} - 3713 x^{8} - 26568 x^{7} - 772 x^{6} + 29700 x^{5} + 347147 x^{4} - 604188 x^{3} - 261314 x^{2} + 228044 x + 1113751 \)

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:  \(24759631762948096000000000000=2^{44}\cdot 5^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.51$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(560=2^{4}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(491,·)$, $\chi_{560}(449,·)$, $\chi_{560}(307,·)$, $\chi_{560}(83,·)$, $\chi_{560}(281,·)$, $\chi_{560}(153,·)$, $\chi_{560}(27,·)$, $\chi_{560}(97,·)$, $\chi_{560}(99,·)$, $\chi_{560}(169,·)$, $\chi_{560}(363,·)$, $\chi_{560}(433,·)$, $\chi_{560}(211,·)$, $\chi_{560}(377,·)$, $\chi_{560}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{33537747966747} a^{14} - \frac{1704166806971}{33537747966747} a^{13} + \frac{5355810705352}{33537747966747} a^{12} + \frac{2362197408109}{11179249322249} a^{11} + \frac{7131673588819}{33537747966747} a^{10} + \frac{10704374354978}{33537747966747} a^{9} - \frac{8481010775291}{33537747966747} a^{8} + \frac{664900662464}{11179249322249} a^{7} + \frac{13460651601848}{33537747966747} a^{6} - \frac{5815603886471}{33537747966747} a^{5} + \frac{38127753047}{360620945879} a^{4} + \frac{3047903221789}{33537747966747} a^{3} - \frac{6988572118226}{33537747966747} a^{2} - \frac{5190680785693}{11179249322249} a - \frac{2926139522542}{33537747966747}$, $\frac{1}{1287309409955507421083882427} a^{15} + \frac{15394097610476}{1287309409955507421083882427} a^{14} + \frac{3947340858806193563966819}{41526109998564755518834917} a^{13} - \frac{477652654218632124462850139}{1287309409955507421083882427} a^{12} - \frac{463815389081976344970405650}{1287309409955507421083882427} a^{11} - \frac{133298068046515760884639415}{429103136651835807027960809} a^{10} + \frac{27807807563564108419531685}{429103136651835807027960809} a^{9} + \frac{20356729992784879809882040}{1287309409955507421083882427} a^{8} + \frac{218214343232252800948750508}{1287309409955507421083882427} a^{7} + \frac{43098938392034786849279362}{429103136651835807027960809} a^{6} + \frac{70318509760388264399924965}{1287309409955507421083882427} a^{5} - \frac{417756621293096209641628100}{1287309409955507421083882427} a^{4} + \frac{353628125814650015395201667}{1287309409955507421083882427} a^{3} - \frac{548744172686627644070105543}{1287309409955507421083882427} a^{2} + \frac{230165561304285222412248047}{1287309409955507421083882427} a - \frac{144739326822610086719390758}{1287309409955507421083882427}$

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

Class group and class number

$C_{2}\times C_{170}$, which has order $340$ (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:  \( 69472.20870795674 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), 4.0.2048.2, \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.51200.2, 4.0.12544000.1, 4.0.12544000.2, 4.4.6125.1, 4.4.392000.1, 8.0.2621440000.1, 8.0.157351936000000.83, 8.8.153664000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
5Data not computed
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$