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Properties

Label 16.0.60300391495...0625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 89^{12}$
Root discriminant $96.89$
Ramified primes $5, 89$
Class number $720$ (GRH)
Class group $[2, 6, 60]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 25600, 176896, 1210688, 8292784, -4940264, 3157636, -1831298, 1160571, -77707, 37414, -2863, 1081, -17, 34, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 34*x^14 - 17*x^13 + 1081*x^12 - 2863*x^11 + 37414*x^10 - 77707*x^9 + 1160571*x^8 - 1831298*x^7 + 3157636*x^6 - 4940264*x^5 + 8292784*x^4 + 1210688*x^3 + 176896*x^2 + 25600*x + 4096)
 
gp: K = bnfinit(x^16 - x^15 + 34*x^14 - 17*x^13 + 1081*x^12 - 2863*x^11 + 37414*x^10 - 77707*x^9 + 1160571*x^8 - 1831298*x^7 + 3157636*x^6 - 4940264*x^5 + 8292784*x^4 + 1210688*x^3 + 176896*x^2 + 25600*x + 4096, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 34 x^{14} - 17 x^{13} + 1081 x^{12} - 2863 x^{11} + 37414 x^{10} - 77707 x^{9} + 1160571 x^{8} - 1831298 x^{7} + 3157636 x^{6} - 4940264 x^{5} + 8292784 x^{4} + 1210688 x^{3} + 176896 x^{2} + 25600 x + 4096 \)

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:  \(60300391495425327222539306640625=5^{12}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.89$
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:  $5, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{10} a^{9} - \frac{1}{10} a^{8} - \frac{2}{5} a^{7} - \frac{1}{10} a^{6} + \frac{1}{10} a^{5} + \frac{3}{10} a^{4} + \frac{1}{5} a^{3} + \frac{3}{10} a^{2} - \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{20} a^{10} - \frac{1}{20} a^{9} - \frac{1}{10} a^{8} - \frac{9}{20} a^{7} - \frac{3}{20} a^{6} - \frac{7}{20} a^{5} - \frac{3}{10} a^{4} - \frac{7}{20} a^{3} - \frac{9}{20} a^{2} + \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{200} a^{11} - \frac{1}{200} a^{10} - \frac{3}{100} a^{9} - \frac{17}{200} a^{8} - \frac{79}{200} a^{7} + \frac{81}{200} a^{6} - \frac{9}{20} a^{5} - \frac{11}{200} a^{4} - \frac{37}{200} a^{3} - \frac{17}{100} a^{2} + \frac{3}{10} a + \frac{7}{25}$, $\frac{1}{400} a^{12} - \frac{1}{400} a^{11} - \frac{3}{200} a^{10} - \frac{17}{400} a^{9} + \frac{1}{400} a^{8} - \frac{39}{400} a^{7} + \frac{3}{8} a^{6} - \frac{11}{400} a^{5} - \frac{157}{400} a^{4} + \frac{83}{200} a^{3} + \frac{1}{20} a^{2} + \frac{1}{25} a + \frac{1}{5}$, $\frac{1}{8132269611217600} a^{13} + \frac{578061263003}{8132269611217600} a^{12} + \frac{5536462654223}{4066134805608800} a^{11} + \frac{8003096370703}{8132269611217600} a^{10} + \frac{386306501958157}{8132269611217600} a^{9} + \frac{238873572116213}{8132269611217600} a^{8} + \frac{380493058215849}{813226961121760} a^{7} + \frac{523205402330289}{1626453922243520} a^{6} + \frac{1683595010242519}{8132269611217600} a^{5} - \frac{1758164938074219}{4066134805608800} a^{4} + \frac{267793837445333}{2033067402804400} a^{3} - \frac{99045648155701}{1016533701402200} a^{2} - \frac{197996309643341}{508266850701100} a + \frac{27865766576864}{127066712675275}$, $\frac{1}{162645392224352000} a^{14} + \frac{7}{162645392224352000} a^{13} - \frac{77809554124043}{81322696112176000} a^{12} - \frac{385309392120497}{162645392224352000} a^{11} + \frac{3382148431151601}{162645392224352000} a^{10} - \frac{7801718442574263}{162645392224352000} a^{9} + \frac{1150656772054447}{81322696112176000} a^{8} - \frac{3948746906175787}{162645392224352000} a^{7} - \frac{1186152050597149}{162645392224352000} a^{6} + \frac{20112329640808691}{81322696112176000} a^{5} + \frac{18754074581621409}{40661348056088000} a^{4} + \frac{2851552657347639}{20330674028044000} a^{3} - \frac{39179099500261}{10165337014022000} a^{2} - \frac{79405251324743}{2541334253505500} a - \frac{78031312795949}{635333563376375}$, $\frac{1}{650581568897408000} a^{15} - \frac{1}{650581568897408000} a^{14} + \frac{9}{325290784448704000} a^{13} - \frac{795636376508929}{650581568897408000} a^{12} + \frac{1385618920217}{650581568897408000} a^{11} - \frac{10180790702755871}{650581568897408000} a^{10} - \frac{14777651349161781}{325290784448704000} a^{9} - \frac{36736688871941019}{650581568897408000} a^{8} - \frac{110893024564486373}{650581568897408000} a^{7} - \frac{70443850557152873}{325290784448704000} a^{6} - \frac{1601296403926479}{32529078444870400} a^{5} - \frac{13662799060153817}{81322696112176000} a^{4} + \frac{7077770961655543}{40661348056088000} a^{3} - \frac{2469386994541821}{10165337014022000} a^{2} - \frac{266266504488999}{1270667126752750} a - \frac{288281784691117}{635333563376375}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{60}$, which has order $720$ (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:  \( \frac{4637747729557}{130116313779481600} a^{15} - \frac{4637259558101}{130116313779481600} a^{14} + \frac{78841711402469}{65058156889740800} a^{13} - \frac{78841711402469}{130116313779481600} a^{12} + \frac{5013405295651117}{130116313779481600} a^{11} - \frac{13277871749721691}{130116313779481600} a^{10} + \frac{86757630684024879}{65058156889740800} a^{9} - \frac{360385462820685799}{130116313779481600} a^{8} + \frac{5382435520239697047}{130116313779481600} a^{7} - \frac{4246549070821137493}{65058156889740800} a^{6} + \frac{3661079797441861813}{32529078444870400} a^{5} - \frac{2865977725733987177}{16264539222435200} a^{4} + \frac{2403740010481663543}{8132269611217600} a^{3} + \frac{87732273800029769}{2033067402804400} a^{2} + \frac{3204683681123887}{508266850701100} a + \frac{4637747729557}{5082668507011} \) (order $10$)
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:  \( 14693160.0683 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{445}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{5}, \sqrt{89})\), 4.4.17624225.1 x2, 4.4.3524845.1 x2, 4.0.990125.2, \(\Q(\zeta_{5})\), 8.8.310613306850625.2, 8.0.980347515625.5, 8.0.7765332671265625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$89$89.8.6.1$x^{8} - 4361 x^{4} + 10265616$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
89.8.6.1$x^{8} - 4361 x^{4} + 10265616$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$