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Properties

Label 16.0.16071660170...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 37^{12}$
Root discriminant $50.16$
Ramified primes $5, 37$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5764801, -823543, -470596, 33614, -57624, -66199, 24255, 3780, -488, 540, 495, -193, -24, 2, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 4*x^14 + 2*x^13 - 24*x^12 - 193*x^11 + 495*x^10 + 540*x^9 - 488*x^8 + 3780*x^7 + 24255*x^6 - 66199*x^5 - 57624*x^4 + 33614*x^3 - 470596*x^2 - 823543*x + 5764801)
 
gp: K = bnfinit(x^16 - x^15 - 4*x^14 + 2*x^13 - 24*x^12 - 193*x^11 + 495*x^10 + 540*x^9 - 488*x^8 + 3780*x^7 + 24255*x^6 - 66199*x^5 - 57624*x^4 + 33614*x^3 - 470596*x^2 - 823543*x + 5764801, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 4 x^{14} + 2 x^{13} - 24 x^{12} - 193 x^{11} + 495 x^{10} + 540 x^{9} - 488 x^{8} + 3780 x^{7} + 24255 x^{6} - 66199 x^{5} - 57624 x^{4} + 33614 x^{3} - 470596 x^{2} - 823543 x + 5764801 \)

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:  \(1607166017050789863525390625=5^{12}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.16$
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:  $5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(185=5\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{185}(1,·)$, $\chi_{185}(68,·)$, $\chi_{185}(6,·)$, $\chi_{185}(73,·)$, $\chi_{185}(142,·)$, $\chi_{185}(147,·)$, $\chi_{185}(149,·)$, $\chi_{185}(154,·)$, $\chi_{185}(31,·)$, $\chi_{185}(36,·)$, $\chi_{185}(38,·)$, $\chi_{185}(43,·)$, $\chi_{185}(112,·)$, $\chi_{185}(179,·)$, $\chi_{185}(117,·)$, $\chi_{185}(184,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{14} a^{9} - \frac{1}{14} a^{8} + \frac{3}{14} a^{7} + \frac{1}{7} a^{6} - \frac{3}{14} a^{5} + \frac{3}{14} a^{4} + \frac{5}{14} a^{3} + \frac{1}{14} a^{2} + \frac{1}{7} a - \frac{1}{2}$, $\frac{1}{196} a^{10} + \frac{3}{98} a^{9} + \frac{19}{98} a^{8} - \frac{13}{98} a^{7} - \frac{5}{98} a^{6} - \frac{9}{98} a^{5} + \frac{13}{98} a^{4} - \frac{31}{98} a^{3} + \frac{29}{98} a^{2} + \frac{5}{14} a + \frac{1}{4}$, $\frac{1}{1372} a^{11} - \frac{1}{1372} a^{10} - \frac{1}{343} a^{9} + \frac{1}{686} a^{8} - \frac{6}{343} a^{7} + \frac{75}{686} a^{6} + \frac{38}{343} a^{5} + \frac{135}{343} a^{4} + \frac{99}{686} a^{3} - \frac{12}{49} a^{2} + \frac{5}{28} a - \frac{1}{4}$, $\frac{1}{28812} a^{12} - \frac{2}{7203} a^{11} + \frac{1}{9604} a^{10} + \frac{5}{4802} a^{9} - \frac{19}{14406} a^{8} + \frac{198}{2401} a^{7} - \frac{739}{7203} a^{6} + \frac{713}{4802} a^{5} + \frac{5069}{14406} a^{4} + \frac{57}{686} a^{3} - \frac{15}{196} a^{2} + \frac{10}{21} a - \frac{5}{12}$, $\frac{1}{129279444} a^{13} + \frac{205}{43093148} a^{12} - \frac{10249}{32319861} a^{11} + \frac{6319}{21546574} a^{10} + \frac{40678}{32319861} a^{9} - \frac{6801355}{32319861} a^{8} + \frac{2719471}{64639722} a^{7} - \frac{4597451}{32319861} a^{6} - \frac{5064755}{32319861} a^{5} + \frac{1777388}{4617123} a^{4} - \frac{98747}{879452} a^{3} - \frac{163943}{376908} a^{2} + \frac{1625}{4487} a + \frac{907}{3846}$, $\frac{1}{904956108} a^{14} - \frac{1}{904956108} a^{13} - \frac{11519}{904956108} a^{12} + \frac{86159}{301652036} a^{11} - \frac{50231}{226239027} a^{10} - \frac{14060377}{452478054} a^{9} + \frac{13940233}{452478054} a^{8} + \frac{15742316}{226239027} a^{7} - \frac{52455014}{226239027} a^{6} + \frac{6170404}{32319861} a^{5} + \frac{753283}{6156164} a^{4} + \frac{360133}{2638356} a^{3} - \frac{4861}{376908} a^{2} - \frac{3593}{7692} a - \frac{269}{641}$, $\frac{1}{12669385512} a^{15} + \frac{1}{2111564252} a^{14} + \frac{19}{6334692756} a^{13} - \frac{92819}{6334692756} a^{12} - \frac{290441}{2111564252} a^{11} - \frac{24439699}{12669385512} a^{10} - \frac{38829823}{3167346378} a^{9} - \frac{85798017}{1055782126} a^{8} + \frac{365610359}{1583673189} a^{7} - \frac{7522108}{75413009} a^{6} - \frac{1473187}{86186296} a^{5} - \frac{264017}{6156164} a^{4} + \frac{5247}{125636} a^{3} - \frac{93743}{376908} a^{2} - \frac{7649}{53844} a + \frac{5021}{15384}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  \( -\frac{12045}{4223128504} a^{15} - \frac{2409}{2111564252} a^{14} - \frac{26499}{2111564252} a^{13} - \frac{64693}{1055782126} a^{12} + \frac{363759}{2111564252} a^{11} + \frac{2493315}{4223128504} a^{10} + \frac{31317}{1055782126} a^{9} + \frac{1982607}{1055782126} a^{8} + \frac{4841232}{527891063} a^{7} - \frac{257763}{10773287} a^{6} - \frac{869649}{12312328} a^{5} - \frac{12045}{879452} a^{4} - \frac{31317}{125636} a^{3} - \frac{62799}{62818} a^{2} + \frac{7227}{2564} a + \frac{16863}{5128} \) (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:  \( 1484842.58736 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{185}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{5}) \), 4.4.6331625.1, \(\Q(\sqrt{5}, \sqrt{37})\), 4.4.6331625.2, 4.0.50653.1, 4.0.1266325.1, \(\Q(\zeta_{5})\), 4.0.171125.1, 8.8.40089475140625.1, 8.0.1603579005625.1, 8.0.29283765625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
37Data not computed