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Properties

Label 16.0.52454638271...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 29^{8}$
Root discriminant $72.03$
Ramified primes $2, 5, 29$
Class number $15360$ (GRH)
Class group $[4, 16, 240]$ (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![94703321, 36634200, 60000708, 14908160, 14613895, 2116800, 1964998, 143120, 190844, 6960, 14572, 80, 770, -20, 32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 32*x^14 - 20*x^13 + 770*x^12 + 80*x^11 + 14572*x^10 + 6960*x^9 + 190844*x^8 + 143120*x^7 + 1964998*x^6 + 2116800*x^5 + 14613895*x^4 + 14908160*x^3 + 60000708*x^2 + 36634200*x + 94703321)
 
gp: K = bnfinit(x^16 + 32*x^14 - 20*x^13 + 770*x^12 + 80*x^11 + 14572*x^10 + 6960*x^9 + 190844*x^8 + 143120*x^7 + 1964998*x^6 + 2116800*x^5 + 14613895*x^4 + 14908160*x^3 + 60000708*x^2 + 36634200*x + 94703321, 1)
 

Normalized defining polynomial

\( x^{16} + 32 x^{14} - 20 x^{13} + 770 x^{12} + 80 x^{11} + 14572 x^{10} + 6960 x^{9} + 190844 x^{8} + 143120 x^{7} + 1964998 x^{6} + 2116800 x^{5} + 14613895 x^{4} + 14908160 x^{3} + 60000708 x^{2} + 36634200 x + 94703321 \)

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:  \(524546382716993536000000000000=2^{32}\cdot 5^{12}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.03$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1160=2^{3}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1160}(1,·)$, $\chi_{1160}(579,·)$, $\chi_{1160}(581,·)$, $\chi_{1160}(1159,·)$, $\chi_{1160}(523,·)$, $\chi_{1160}(1103,·)$, $\chi_{1160}(407,·)$, $\chi_{1160}(987,·)$, $\chi_{1160}(349,·)$, $\chi_{1160}(929,·)$, $\chi_{1160}(231,·)$, $\chi_{1160}(811,·)$, $\chi_{1160}(173,·)$, $\chi_{1160}(753,·)$, $\chi_{1160}(57,·)$, $\chi_{1160}(637,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{1353} a^{14} + \frac{10}{1353} a^{13} - \frac{95}{1353} a^{12} + \frac{368}{1353} a^{11} - \frac{13}{123} a^{10} - \frac{183}{451} a^{9} + \frac{17}{451} a^{8} - \frac{167}{451} a^{7} - \frac{280}{1353} a^{6} - \frac{317}{1353} a^{5} - \frac{53}{1353} a^{4} - \frac{158}{451} a^{3} + \frac{217}{1353} a^{2} - \frac{124}{451} a - \frac{544}{1353}$, $\frac{1}{14562958817682548360588440708899381795093} a^{15} + \frac{1448388805612017972403177999042044242}{14562958817682548360588440708899381795093} a^{14} - \frac{1185882480101102933666493785180958231988}{14562958817682548360588440708899381795093} a^{13} - \frac{702654990271262622186162831445975150907}{14562958817682548360588440708899381795093} a^{12} - \frac{2127495646323370931813254366248752394991}{4854319605894182786862813569633127265031} a^{11} + \frac{1152586260751022159571046664637149517564}{4854319605894182786862813569633127265031} a^{10} - \frac{865896640089728556753048158047307331371}{4854319605894182786862813569633127265031} a^{9} + \frac{1681051048762922127250596906442998095475}{4854319605894182786862813569633127265031} a^{8} + \frac{554400889868752886073049066911035187512}{14562958817682548360588440708899381795093} a^{7} + \frac{2025978063989154443440846704331859664178}{4854319605894182786862813569633127265031} a^{6} - \frac{6968385205154473605414707490981163314394}{14562958817682548360588440708899381795093} a^{5} - \frac{1546618700076252002470628428612418562430}{4854319605894182786862813569633127265031} a^{4} - \frac{1252720607703590671630716488237645875930}{4854319605894182786862813569633127265031} a^{3} + \frac{3475933412431640211734740002359813803952}{14562958817682548360588440708899381795093} a^{2} + \frac{1313736143930597627056630796235793805252}{4854319605894182786862813569633127265031} a - \frac{1006166134389883224186597729755941880915}{4854319605894182786862813569633127265031}$

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

Class group and class number

$C_{4}\times C_{16}\times C_{240}$, which has order $15360$ (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:  \( 7114.135357253273 \) (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

\(\Q(\sqrt{-290}) \), \(\Q(\sqrt{-145}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-58}) \), \(\Q(\sqrt{-29}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{-145})\), \(\Q(\sqrt{5}, \sqrt{-58})\), \(\Q(\sqrt{10}, \sqrt{-29})\), \(\Q(\sqrt{5}, \sqrt{-29})\), \(\Q(\sqrt{10}, \sqrt{-58})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-29})\), 4.0.6728000.5, \(\Q(\zeta_{20})^+\), 4.4.8000.1, 4.0.105125.2, 8.0.28970229760000.62, 8.0.724255744000000.55, 8.0.45265984000000.50, 8.0.724255744000000.64, 8.0.2829124000000.1, 8.0.45265984000000.54, \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$