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Properties

Label 16.0.11684193984...000.14
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}$
Root discriminant $100.98$
Ramified primes $2, 3, 5, 19$
Class number $204800$ (GRH)
Class group $[2, 4, 4, 80, 80]$ (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![31810900, -15996200, 19889100, -8321000, 5551070, -1819760, 870290, -250700, 95421, -24808, 7958, -1676, 340, -56, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 340*x^12 - 1676*x^11 + 7958*x^10 - 24808*x^9 + 95421*x^8 - 250700*x^7 + 870290*x^6 - 1819760*x^5 + 5551070*x^4 - 8321000*x^3 + 19889100*x^2 - 15996200*x + 31810900)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 340*x^12 - 1676*x^11 + 7958*x^10 - 24808*x^9 + 95421*x^8 - 250700*x^7 + 870290*x^6 - 1819760*x^5 + 5551070*x^4 - 8321000*x^3 + 19889100*x^2 - 15996200*x + 31810900, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 340 x^{12} - 1676 x^{11} + 7958 x^{10} - 24808 x^{9} + 95421 x^{8} - 250700 x^{7} + 870290 x^{6} - 1819760 x^{5} + 5551070 x^{4} - 8321000 x^{3} + 19889100 x^{2} - 15996200 x + 31810900 \)

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:  \(116841939847873560576000000000000=2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.98$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2280=2^{3}\cdot 3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2280}(1,·)$, $\chi_{2280}(1027,·)$, $\chi_{2280}(1483,·)$, $\chi_{2280}(77,·)$, $\chi_{2280}(911,·)$, $\chi_{2280}(721,·)$, $\chi_{2280}(1747,·)$, $\chi_{2280}(533,·)$, $\chi_{2280}(1559,·)$, $\chi_{2280}(1369,·)$, $\chi_{2280}(2203,·)$, $\chi_{2280}(797,·)$, $\chi_{2280}(1253,·)$, $\chi_{2280}(2279,·)$, $\chi_{2280}(2089,·)$, $\chi_{2280}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{10} a^{4}$, $\frac{1}{110} a^{9} + \frac{1}{110} a^{8} + \frac{18}{55} a^{7} + \frac{8}{55} a^{6} + \frac{1}{10} a^{5} - \frac{9}{22} a^{4} + \frac{1}{11} a^{3} + \frac{3}{11} a^{2} + \frac{5}{11} a$, $\frac{1}{110} a^{10} + \frac{1}{55} a^{8} + \frac{1}{55} a^{7} + \frac{17}{110} a^{6} - \frac{17}{55} a^{5} + \frac{1}{5} a^{4} + \frac{2}{11} a^{3} + \frac{2}{11} a^{2} - \frac{5}{11} a$, $\frac{1}{110} a^{11} - \frac{1}{2} a^{7} + \frac{2}{5} a^{6} + \frac{1}{11} a$, $\frac{1}{110} a^{12} + \frac{2}{5} a^{7} - \frac{1}{2} a^{4} + \frac{1}{11} a^{2}$, $\frac{1}{6490} a^{13} + \frac{23}{6490} a^{12} - \frac{3}{1298} a^{11} - \frac{23}{6490} a^{10} + \frac{23}{6490} a^{9} + \frac{13}{1298} a^{8} + \frac{2113}{6490} a^{7} + \frac{2441}{6490} a^{6} + \frac{1172}{3245} a^{5} + \frac{1589}{3245} a^{4} - \frac{21}{59} a^{3} + \frac{277}{649} a^{2} - \frac{192}{649} a - \frac{10}{59}$, $\frac{1}{120803791647021010} a^{14} - \frac{7}{120803791647021010} a^{13} + \frac{295251146326839}{120803791647021010} a^{12} + \frac{424925697439439}{120803791647021010} a^{11} + \frac{448652633417711}{120803791647021010} a^{10} - \frac{140630929607947}{60401895823510505} a^{9} - \frac{320100741356999}{12080379164702101} a^{8} - \frac{57709356657603931}{120803791647021010} a^{7} + \frac{10584533670018727}{60401895823510505} a^{6} - \frac{346970961092307}{2047521892322390} a^{5} - \frac{912209668165009}{120803791647021010} a^{4} - \frac{1860079061314714}{12080379164702101} a^{3} + \frac{3541331749054973}{12080379164702101} a^{2} + \frac{394769987348881}{1098216287700191} a + \frac{23335868061830}{99837844336381}$, $\frac{1}{252359120750626889890} a^{15} + \frac{1037}{252359120750626889890} a^{14} - \frac{48310842481591}{2498605155946800890} a^{13} + \frac{170052945213750208}{126179560375313444945} a^{12} - \frac{148221161182521153}{252359120750626889890} a^{11} - \frac{32605762007476979}{126179560375313444945} a^{10} + \frac{76510392571825253}{126179560375313444945} a^{9} + \frac{2167439198240807869}{126179560375313444945} a^{8} + \frac{11467970590416174833}{126179560375313444945} a^{7} - \frac{4296727281497765149}{50471824150125377978} a^{6} + \frac{95008767245556283089}{252359120750626889890} a^{5} - \frac{8364245014568341243}{126179560375313444945} a^{4} + \frac{4404301812925100056}{25235912075062688989} a^{3} + \frac{3980235000760845778}{25235912075062688989} a^{2} - \frac{1583357977668683}{4017817556927669} a + \frac{100251253120172193}{208561256818699909}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{80}\times C_{80}$, which has order $204800$ (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:  \( 54685.79274709562 \) (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

\(\Q(\sqrt{-285}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-57}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{3}, \sqrt{-95})\), \(\Q(\sqrt{5}, \sqrt{-57})\), \(\Q(\sqrt{15}, \sqrt{-19})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-19})\), \(\Q(\sqrt{5}, \sqrt{-19})\), \(\Q(\sqrt{15}, \sqrt{-57})\), 4.0.2888000.1, 4.4.72000.1, 4.0.25992000.5, 4.4.8000.1, 8.0.1688960160000.4, 8.0.10809345024000000.231, 8.0.10809345024000000.253, 8.0.10809345024000000.181, 8.8.82944000000.1, 8.0.8340544000000.3, 8.0.675584064000000.148

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.2$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 20$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.2$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 20$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
$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}$