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Properties

Label 16.0.53902309800...8304.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 43^{8}$
Root discriminant $96.21$
Ramified primes $2, 43$
Class number $116125$ (GRH)
Class group $[5, 23225]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![745835231, -343425912, 418600684, -160536640, 106670842, -34360072, 16096672, -4351384, 1570907, -351880, 101264, -18200, 4186, -560, 100, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 100*x^14 - 560*x^13 + 4186*x^12 - 18200*x^11 + 101264*x^10 - 351880*x^9 + 1570907*x^8 - 4351384*x^7 + 16096672*x^6 - 34360072*x^5 + 106670842*x^4 - 160536640*x^3 + 418600684*x^2 - 343425912*x + 745835231)
 
gp: K = bnfinit(x^16 - 8*x^15 + 100*x^14 - 560*x^13 + 4186*x^12 - 18200*x^11 + 101264*x^10 - 351880*x^9 + 1570907*x^8 - 4351384*x^7 + 16096672*x^6 - 34360072*x^5 + 106670842*x^4 - 160536640*x^3 + 418600684*x^2 - 343425912*x + 745835231, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 100 x^{14} - 560 x^{13} + 4186 x^{12} - 18200 x^{11} + 101264 x^{10} - 351880 x^{9} + 1570907 x^{8} - 4351384 x^{7} + 16096672 x^{6} - 34360072 x^{5} + 106670842 x^{4} - 160536640 x^{3} + 418600684 x^{2} - 343425912 x + 745835231 \)

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:  \(53902309800791645700994109538304=2^{62}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.21$
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:  $2, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1376=2^{5}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{1376}(1,·)$, $\chi_{1376}(517,·)$, $\chi_{1376}(257,·)$, $\chi_{1376}(1033,·)$, $\chi_{1376}(429,·)$, $\chi_{1376}(85,·)$, $\chi_{1376}(345,·)$, $\chi_{1376}(601,·)$, $\chi_{1376}(861,·)$, $\chi_{1376}(773,·)$, $\chi_{1376}(945,·)$, $\chi_{1376}(173,·)$, $\chi_{1376}(1117,·)$, $\chi_{1376}(689,·)$, $\chi_{1376}(1205,·)$, $\chi_{1376}(1289,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{111480229810044983441} a^{14} - \frac{7}{111480229810044983441} a^{13} + \frac{46746984994961028571}{111480229810044983441} a^{12} + \frac{53958779460368778988}{111480229810044983441} a^{11} - \frac{267506639978536504}{111480229810044983441} a^{10} + \frac{8376422291714633781}{111480229810044983441} a^{9} - \frac{42050536357028640352}{111480229810044983441} a^{8} + \frac{42608806881613427230}{111480229810044983441} a^{7} + \frac{50329191989710438221}{111480229810044983441} a^{6} + \frac{44505860967730948616}{111480229810044983441} a^{5} + \frac{18243665976193567143}{111480229810044983441} a^{4} - \frac{3893509374880666736}{111480229810044983441} a^{3} + \frac{6253175469858645509}{111480229810044983441} a^{2} - \frac{1850876040173657579}{111480229810044983441} a + \frac{47933892199668067882}{111480229810044983441}$, $\frac{1}{735685795093709546928035809} a^{15} + \frac{3299617}{735685795093709546928035809} a^{14} - \frac{228542339459923734309943108}{735685795093709546928035809} a^{13} - \frac{199441784254263808037520532}{735685795093709546928035809} a^{12} + \frac{1339492625165610028171256}{735685795093709546928035809} a^{11} + \frac{263536402291596969044999657}{735685795093709546928035809} a^{10} + \frac{293499283632931727156123612}{735685795093709546928035809} a^{9} + \frac{225855654367636873792833536}{735685795093709546928035809} a^{8} + \frac{232198626566638976713938261}{735685795093709546928035809} a^{7} - \frac{132574257816896782325014796}{735685795093709546928035809} a^{6} + \frac{257824059274646177926262803}{735685795093709546928035809} a^{5} - \frac{2460705797879876543314179}{23731799841732566029936639} a^{4} + \frac{19555398431492469512548982}{735685795093709546928035809} a^{3} - \frac{325865822481857642801564728}{735685795093709546928035809} a^{2} + \frac{100263609367871426667566233}{735685795093709546928035809} a + \frac{10010436168511528285504761}{23731799841732566029936639}$

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

Class group and class number

$C_{5}\times C_{23225}$, which has order $116125$ (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:  \( 15753.94986242651 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

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 $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-86}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{2}, \sqrt{-43})\), \(\Q(\zeta_{16})^+\), 4.0.3786752.2, 8.0.14339490709504.11, \(\Q(\zeta_{32})^+\), 8.0.7341819243266048.8

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
43Data not computed