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Properties

Label 16.16.1048315135...4369.1
Degree $16$
Signature $[16, 0]$
Discriminant $17^{14}\cdot 53^{8}$
Root discriminant $86.85$
Ramified primes $17, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-169, -676, 22473, -12992, -328065, 706524, -131564, -491508, 120123, 109926, -27184, -9898, 2466, 344, -86, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 86*x^14 + 344*x^13 + 2466*x^12 - 9898*x^11 - 27184*x^10 + 109926*x^9 + 120123*x^8 - 491508*x^7 - 131564*x^6 + 706524*x^5 - 328065*x^4 - 12992*x^3 + 22473*x^2 - 676*x - 169)
 
gp: K = bnfinit(x^16 - 4*x^15 - 86*x^14 + 344*x^13 + 2466*x^12 - 9898*x^11 - 27184*x^10 + 109926*x^9 + 120123*x^8 - 491508*x^7 - 131564*x^6 + 706524*x^5 - 328065*x^4 - 12992*x^3 + 22473*x^2 - 676*x - 169, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 86 x^{14} + 344 x^{13} + 2466 x^{12} - 9898 x^{11} - 27184 x^{10} + 109926 x^{9} + 120123 x^{8} - 491508 x^{7} - 131564 x^{6} + 706524 x^{5} - 328065 x^{4} - 12992 x^{3} + 22473 x^{2} - 676 x - 169 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10483151353726139536553735554369=17^{14}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.85$
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:  $17, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{13} a^{7} - \frac{2}{13} a^{6} + \frac{2}{13} a^{5} + \frac{5}{13} a^{4} + \frac{1}{13} a^{3} + \frac{6}{13} a^{2} + \frac{5}{13} a$, $\frac{1}{442} a^{8} - \frac{1}{221} a^{7} + \frac{105}{221} a^{6} - \frac{95}{221} a^{5} + \frac{53}{442} a^{4} - \frac{75}{221} a^{3} + \frac{57}{442} a^{2} - \frac{9}{34} a - \frac{13}{34}$, $\frac{1}{442} a^{9} + \frac{1}{221} a^{7} + \frac{98}{221} a^{6} + \frac{149}{442} a^{5} - \frac{90}{221} a^{4} - \frac{5}{442} a^{3} + \frac{99}{442} a^{2} - \frac{97}{442} a + \frac{4}{17}$, $\frac{1}{442} a^{10} - \frac{2}{221} a^{7} + \frac{137}{442} a^{6} - \frac{8}{17} a^{5} + \frac{15}{34} a^{4} + \frac{15}{34} a^{3} - \frac{109}{442} a^{2} + \frac{101}{221} a - \frac{4}{17}$, $\frac{1}{442} a^{11} - \frac{7}{442} a^{7} + \frac{10}{221} a^{6} + \frac{47}{442} a^{5} + \frac{13}{34} a^{4} + \frac{3}{34} a^{3} + \frac{28}{221} a^{2} + \frac{37}{221} a + \frac{8}{17}$, $\frac{1}{442} a^{12} + \frac{3}{221} a^{7} + \frac{191}{442} a^{6} + \frac{165}{442} a^{5} - \frac{16}{221} a^{4} - \frac{55}{221} a^{3} + \frac{31}{442} a^{2} - \frac{13}{34} a + \frac{11}{34}$, $\frac{1}{5746} a^{13} - \frac{1}{5746} a^{12} + \frac{5}{5746} a^{11} + \frac{2}{2873} a^{10} - \frac{1}{2873} a^{9} - \frac{2}{2873} a^{8} - \frac{44}{2873} a^{7} - \frac{41}{169} a^{6} - \frac{497}{2873} a^{5} + \frac{63}{338} a^{4} + \frac{89}{2873} a^{3} - \frac{171}{2873} a^{2} - \frac{4}{221} a + \frac{9}{34}$, $\frac{1}{511394} a^{14} + \frac{43}{511394} a^{13} + \frac{3}{3026} a^{12} - \frac{148}{255697} a^{11} + \frac{499}{511394} a^{10} - \frac{23}{30082} a^{9} + \frac{503}{511394} a^{8} - \frac{9731}{255697} a^{7} + \frac{70257}{511394} a^{6} - \frac{54684}{255697} a^{5} - \frac{743}{2873} a^{4} + \frac{87660}{255697} a^{3} + \frac{409}{511394} a^{2} - \frac{2571}{39338} a - \frac{423}{3026}$, $\frac{1}{31859099053366} a^{15} + \frac{20560273}{31859099053366} a^{14} - \frac{2326697277}{31859099053366} a^{13} - \frac{12446963123}{15929549526683} a^{12} + \frac{9691500035}{31859099053366} a^{11} - \frac{12572040814}{15929549526683} a^{10} + \frac{481238811}{1225349963591} a^{9} + \frac{28898159371}{31859099053366} a^{8} + \frac{375137682121}{31859099053366} a^{7} - \frac{3953607904109}{31859099053366} a^{6} + \frac{14073182416451}{31859099053366} a^{5} + \frac{1339853919464}{15929549526683} a^{4} - \frac{3466465045905}{31859099053366} a^{3} - \frac{6110352578019}{15929549526683} a^{2} - \frac{880207775853}{2450699927182} a - \frac{52030826577}{188515379014}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 31959664273.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

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 $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{901}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{17}, \sqrt{53})\), 4.4.4913.1, 4.4.13800617.2, 8.8.190457029580689.3, 8.8.1152641332457.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
$53$53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$