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Properties

Label 16.0.12255966294...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 43^{8}$
Root discriminant $87.70$
Ramified primes $2, 5, 43$
Class number $189000$ (GRH)
Class group $[3, 30, 2100]$ (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![738052316, -345252216, 421058284, -161864696, 107501834, -34569744, 16182278, -4365228, 1574821, -352160, 101298, -18188, 4184, -560, 100, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 100*x^14 - 560*x^13 + 4184*x^12 - 18188*x^11 + 101298*x^10 - 352160*x^9 + 1574821*x^8 - 4365228*x^7 + 16182278*x^6 - 34569744*x^5 + 107501834*x^4 - 161864696*x^3 + 421058284*x^2 - 345252216*x + 738052316)
 
gp: K = bnfinit(x^16 - 8*x^15 + 100*x^14 - 560*x^13 + 4184*x^12 - 18188*x^11 + 101298*x^10 - 352160*x^9 + 1574821*x^8 - 4365228*x^7 + 16182278*x^6 - 34569744*x^5 + 107501834*x^4 - 161864696*x^3 + 421058284*x^2 - 345252216*x + 738052316, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 100 x^{14} - 560 x^{13} + 4184 x^{12} - 18188 x^{11} + 101298 x^{10} - 352160 x^{9} + 1574821 x^{8} - 4365228 x^{7} + 16182278 x^{6} - 34569744 x^{5} + 107501834 x^{4} - 161864696 x^{3} + 421058284 x^{2} - 345252216 x + 738052316 \)

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:  \(12255966294285746176000000000000=2^{32}\cdot 5^{12}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.70$
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:  $2, 5, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1720=2^{3}\cdot 5\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{1720}(1,·)$, $\chi_{1720}(1289,·)$, $\chi_{1720}(1547,·)$, $\chi_{1720}(1549,·)$, $\chi_{1720}(343,·)$, $\chi_{1720}(601,·)$, $\chi_{1720}(603,·)$, $\chi_{1720}(861,·)$, $\chi_{1720}(1203,·)$, $\chi_{1720}(429,·)$, $\chi_{1720}(687,·)$, $\chi_{1720}(689,·)$, $\chi_{1720}(947,·)$, $\chi_{1720}(1461,·)$, $\chi_{1720}(1463,·)$, $\chi_{1720}(87,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{38} a^{11} + \frac{2}{19} a^{10} + \frac{7}{38} a^{9} + \frac{5}{38} a^{8} + \frac{13}{38} a^{7} - \frac{3}{19} a^{6} - \frac{15}{38} a^{5} + \frac{3}{38} a^{4} - \frac{1}{19} a^{3} + \frac{3}{19} a^{2} - \frac{2}{19} a - \frac{3}{19}$, $\frac{1}{38} a^{12} - \frac{9}{38} a^{10} - \frac{2}{19} a^{9} - \frac{7}{38} a^{8} + \frac{9}{19} a^{7} + \frac{9}{38} a^{6} + \frac{3}{19} a^{5} - \frac{7}{19} a^{4} + \frac{7}{19} a^{3} + \frac{5}{19} a^{2} + \frac{5}{19} a - \frac{7}{19}$, $\frac{1}{38} a^{13} - \frac{3}{19} a^{10} - \frac{1}{38} a^{9} + \frac{3}{19} a^{8} + \frac{6}{19} a^{7} - \frac{5}{19} a^{6} - \frac{8}{19} a^{5} - \frac{8}{19} a^{4} - \frac{4}{19} a^{3} - \frac{6}{19} a^{2} - \frac{6}{19} a - \frac{8}{19}$, $\frac{1}{733789229840934542} a^{14} - \frac{7}{733789229840934542} a^{13} + \frac{56439373306182}{19310242890550909} a^{12} + \frac{3221032888370752}{366894614920467271} a^{11} + \frac{144459234088059261}{733789229840934542} a^{10} - \frac{82961322185502383}{733789229840934542} a^{9} + \frac{17467195758858364}{366894614920467271} a^{8} + \frac{98585087949846654}{366894614920467271} a^{7} + \frac{24420775235517867}{366894614920467271} a^{6} + \frac{48863741421159044}{366894614920467271} a^{5} - \frac{53227414643409284}{366894614920467271} a^{4} + \frac{95881377585451035}{366894614920467271} a^{3} + \frac{163909236748122928}{366894614920467271} a^{2} + \frac{51813735275759471}{366894614920467271} a + \frac{170300190780741743}{366894614920467271}$, $\frac{1}{253288632567263944273018} a^{15} + \frac{86291}{126644316283631972136509} a^{14} - \frac{444635508101640803841}{253288632567263944273018} a^{13} + \frac{1630141422495647353620}{126644316283631972136509} a^{12} + \frac{55657105200617569911}{11513119662148361103319} a^{11} - \frac{39139799635246743336743}{253288632567263944273018} a^{10} - \frac{1366362806701240060947}{23026239324296722206638} a^{9} - \frac{35429875332921039026}{11513119662148361103319} a^{8} + \frac{4715866384030971328149}{253288632567263944273018} a^{7} - \frac{63227899299886864803781}{253288632567263944273018} a^{6} + \frac{1856009899528339170233}{126644316283631972136509} a^{5} - \frac{17317971596453438897868}{126644316283631972136509} a^{4} + \frac{25026744710800563315355}{126644316283631972136509} a^{3} + \frac{58315425947276309178704}{126644316283631972136509} a^{2} - \frac{33283790860298681562680}{126644316283631972136509} a + \frac{1206032816588093041708}{3088885763015413954549}$

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

Class group and class number

$C_{3}\times C_{30}\times C_{2100}$, which has order $189000$ (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

\(\Q(\sqrt{-43}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-86}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-215}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-430}) \), \(\Q(\sqrt{2}, \sqrt{-43})\), \(\Q(\sqrt{5}, \sqrt{-43})\), \(\Q(\sqrt{10}, \sqrt{-43})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-215})\), \(\Q(\sqrt{5}, \sqrt{-86})\), \(\Q(\sqrt{10}, \sqrt{-86})\), 4.0.3698000.1, \(\Q(\zeta_{20})^+\), 4.0.14792000.2, 4.4.8000.1, 8.0.8752130560000.17, 8.0.13675204000000.1, 8.0.218803264000000.15, 8.0.3500852224000000.23, \(\Q(\zeta_{40})^+\), 8.0.3500852224000000.29, 8.0.3500852224000000.15

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$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}$
$43$43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$