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Properties

Label 16.0.28493690558...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{8}\cdot 19^{8}$
Root discriminant $38.99$
Ramified primes $2, 5, 19$
Class number $24$ (GRH)
Class group $[24]$ (GRH)
Galois group $C_2^4$ (as 16T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![556804, -508136, 801260, -714504, 716170, -544768, 393078, -226316, 118149, -49568, 19026, -5740, 1624, -336, 68, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 68*x^14 - 336*x^13 + 1624*x^12 - 5740*x^11 + 19026*x^10 - 49568*x^9 + 118149*x^8 - 226316*x^7 + 393078*x^6 - 544768*x^5 + 716170*x^4 - 714504*x^3 + 801260*x^2 - 508136*x + 556804)
 
gp: K = bnfinit(x^16 - 8*x^15 + 68*x^14 - 336*x^13 + 1624*x^12 - 5740*x^11 + 19026*x^10 - 49568*x^9 + 118149*x^8 - 226316*x^7 + 393078*x^6 - 544768*x^5 + 716170*x^4 - 714504*x^3 + 801260*x^2 - 508136*x + 556804, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 68 x^{14} - 336 x^{13} + 1624 x^{12} - 5740 x^{11} + 19026 x^{10} - 49568 x^{9} + 118149 x^{8} - 226316 x^{7} + 393078 x^{6} - 544768 x^{5} + 716170 x^{4} - 714504 x^{3} + 801260 x^{2} - 508136 x + 556804 \)

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:  \(28493690558847385600000000=2^{32}\cdot 5^{8}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.99$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(760=2^{3}\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{760}(1,·)$, $\chi_{760}(721,·)$, $\chi_{760}(531,·)$, $\chi_{760}(341,·)$, $\chi_{760}(151,·)$, $\chi_{760}(609,·)$, $\chi_{760}(419,·)$, $\chi_{760}(229,·)$, $\chi_{760}(39,·)$, $\chi_{760}(379,·)$, $\chi_{760}(189,·)$, $\chi_{760}(759,·)$, $\chi_{760}(569,·)$, $\chi_{760}(571,·)$, $\chi_{760}(381,·)$, $\chi_{760}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{4} + \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{4}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{5}$, $\frac{1}{826073265096702} a^{14} - \frac{1}{118010466442386} a^{13} - \frac{8228757321983}{275357755032234} a^{12} + \frac{5219377139834}{413036632548351} a^{11} - \frac{1400799605147}{413036632548351} a^{10} + \frac{4347778622}{108665254551} a^{9} + \frac{65448273536413}{826073265096702} a^{8} - \frac{34959960271724}{413036632548351} a^{7} + \frac{56581200229723}{826073265096702} a^{6} + \frac{1434666682517}{39336822147462} a^{5} + \frac{179027919455614}{413036632548351} a^{4} - \frac{15257732439217}{413036632548351} a^{3} - \frac{66028759749020}{137678877516117} a^{2} - \frac{10957833191}{305726597001} a + \frac{150212833114550}{413036632548351}$, $\frac{1}{47088654330307304106} a^{15} + \frac{1583}{2616036351683739117} a^{14} + \frac{2593707687253962707}{47088654330307304106} a^{13} + \frac{1028788445019246827}{23544327165153652053} a^{12} - \frac{805700764619890505}{15696218110102434702} a^{11} - \frac{894067419749742293}{23544327165153652053} a^{10} + \frac{560873054075903771}{23544327165153652053} a^{9} - \frac{559458470194673038}{7848109055051217351} a^{8} + \frac{6124836247152386}{40663777487312007} a^{7} + \frac{5539587588000420383}{23544327165153652053} a^{6} + \frac{3892652580515790563}{47088654330307304106} a^{5} + \frac{3696291582650099558}{7848109055051217351} a^{4} - \frac{2360910857224815373}{23544327165153652053} a^{3} + \frac{7454473346805434449}{23544327165153652053} a^{2} - \frac{1150926669995503582}{7848109055051217351} a - \frac{3228340738641296098}{23544327165153652053}$

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

Class group and class number

$C_{24}$, which has order $24$ (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:  \( -\frac{807519836177}{121991332461936021} a^{15} + \frac{4037599180885}{81327554974624014} a^{14} - \frac{92064661402997}{243982664923872042} a^{13} + \frac{414709536389213}{243982664923872042} a^{12} - \frac{27942246444253}{3872740713077334} a^{11} + \frac{2762173585329455}{121991332461936021} a^{10} - \frac{7560521372615390}{121991332461936021} a^{9} + \frac{1206990033322921}{9036394997180446} a^{8} - \frac{18889543465809743}{81327554974624014} a^{7} + \frac{77018773620383705}{243982664923872042} a^{6} - \frac{76830654175885829}{243982664923872042} a^{5} + \frac{8978748528595910}{40663777487312007} a^{4} - \frac{5005787302800692}{17427333208848003} a^{3} + \frac{37262289787529900}{121991332461936021} a^{2} - \frac{15370753510540921}{40663777487312007} a + \frac{17206041769183933}{121991332461936021} \) (order $8$)
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:  \( 498581.960646 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4$ (as 16T3):

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 $C_2^4$
Character table for $C_2^4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{190}) \), \(\Q(\sqrt{-190}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{95}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{19}) \), \(\Q(\sqrt{38}) \), \(\Q(\sqrt{-38}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{190})\), \(\Q(i, \sqrt{95})\), \(\Q(\sqrt{-2}, \sqrt{-95})\), \(\Q(\sqrt{-2}, \sqrt{95})\), \(\Q(\sqrt{2}, \sqrt{95})\), \(\Q(\sqrt{2}, \sqrt{-95})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{19})\), \(\Q(i, \sqrt{38})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-19})\), \(\Q(\sqrt{-2}, \sqrt{19})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-19})\), \(\Q(\sqrt{2}, \sqrt{19})\), \(\Q(\sqrt{-10}, \sqrt{-19})\), \(\Q(\sqrt{10}, \sqrt{19})\), \(\Q(\sqrt{5}, \sqrt{38})\), \(\Q(\sqrt{-5}, \sqrt{-38})\), \(\Q(\sqrt{-10}, \sqrt{19})\), \(\Q(\sqrt{10}, \sqrt{-19})\), \(\Q(\sqrt{5}, \sqrt{-38})\), \(\Q(\sqrt{-5}, \sqrt{38})\), \(\Q(\sqrt{-10}, \sqrt{38})\), \(\Q(\sqrt{10}, \sqrt{-38})\), \(\Q(\sqrt{5}, \sqrt{-19})\), \(\Q(\sqrt{-5}, \sqrt{19})\), \(\Q(\sqrt{-10}, \sqrt{-38})\), \(\Q(\sqrt{10}, \sqrt{38})\), \(\Q(\sqrt{5}, \sqrt{19})\), \(\Q(\sqrt{-5}, \sqrt{-19})\), 8.0.5337948160000.9, 8.0.40960000.1, 8.0.8540717056.1, 8.0.5337948160000.4, 8.0.5337948160000.8, 8.0.5337948160000.3, 8.0.20851360000.1, 8.0.333621760000.1, 8.0.5337948160000.6, 8.0.5337948160000.7, 8.0.5337948160000.5, 8.0.5337948160000.1, 8.8.5337948160000.1, 8.0.5337948160000.2, 8.0.333621760000.2

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$