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Properties

Label 16.0.22856234040...7953.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 41^{8}$
Root discriminant $91.19$
Ramified primes $17, 41$
Class number $220354$ (GRH)
Class group $[220354]$ (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4576004071, -2876004071, 2876004071, -836004071, 836004071, -122004071, 122004071, -9804071, 9804071, -454071, 454071, -12071, 12071, -171, 171, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 171*x^14 - 171*x^13 + 12071*x^12 - 12071*x^11 + 454071*x^10 - 454071*x^9 + 9804071*x^8 - 9804071*x^7 + 122004071*x^6 - 122004071*x^5 + 836004071*x^4 - 836004071*x^3 + 2876004071*x^2 - 2876004071*x + 4576004071)
 
gp: K = bnfinit(x^16 - x^15 + 171*x^14 - 171*x^13 + 12071*x^12 - 12071*x^11 + 454071*x^10 - 454071*x^9 + 9804071*x^8 - 9804071*x^7 + 122004071*x^6 - 122004071*x^5 + 836004071*x^4 - 836004071*x^3 + 2876004071*x^2 - 2876004071*x + 4576004071, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 171 x^{14} - 171 x^{13} + 12071 x^{12} - 12071 x^{11} + 454071 x^{10} - 454071 x^{9} + 9804071 x^{8} - 9804071 x^{7} + 122004071 x^{6} - 122004071 x^{5} + 836004071 x^{4} - 836004071 x^{3} + 2876004071 x^{2} - 2876004071 x + 4576004071 \)

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:  \(22856234040418247855901029307953=17^{15}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.19$
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:  $17, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(697=17\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{697}(1,·)$, $\chi_{697}(206,·)$, $\chi_{697}(83,·)$, $\chi_{697}(532,·)$, $\chi_{697}(286,·)$, $\chi_{697}(288,·)$, $\chi_{697}(163,·)$, $\chi_{697}(40,·)$, $\chi_{697}(42,·)$, $\chi_{697}(368,·)$, $\chi_{697}(616,·)$, $\chi_{697}(370,·)$, $\chi_{697}(245,·)$, $\chi_{697}(247,·)$, $\chi_{697}(122,·)$, $\chi_{697}(573,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{721445251} a^{9} + \frac{148964684}{721445251} a^{8} + \frac{90}{721445251} a^{7} - \frac{347394547}{721445251} a^{6} + \frac{2700}{721445251} a^{5} - \frac{27520663}{721445251} a^{4} + \frac{30000}{721445251} a^{3} - \frac{220165304}{721445251} a^{2} + \frac{90000}{721445251} a - \frac{275206630}{721445251}$, $\frac{1}{721445251} a^{10} + \frac{100}{721445251} a^{8} - \frac{46756338}{721445251} a^{7} + \frac{3500}{721445251} a^{6} + \frac{334282595}{721445251} a^{5} + \frac{50000}{721445251} a^{4} + \frac{192644641}{721445251} a^{3} + \frac{250000}{721445251} a^{2} + \frac{241777954}{721445251} a + \frac{200000}{721445251}$, $\frac{1}{721445251} a^{11} + \frac{207125533}{721445251} a^{8} - \frac{5500}{721445251} a^{7} - \frac{277080004}{721445251} a^{6} - \frac{220000}{721445251} a^{5} + \frac{58929937}{721445251} a^{4} - \frac{2750000}{721445251} a^{3} - \frac{106494427}{721445251} a^{2} - \frac{8800000}{721445251} a + \frac{105743462}{721445251}$, $\frac{1}{721445251} a^{12} - \frac{6600}{721445251} a^{8} - \frac{160801448}{721445251} a^{7} - \frac{308000}{721445251} a^{6} - \frac{59939638}{721445251} a^{5} - \frac{4950000}{721445251} a^{4} - \frac{64537564}{721445251} a^{3} - \frac{26400000}{721445251} a^{2} + \frac{231614051}{721445251} a - \frac{22000000}{721445251}$, $\frac{1}{721445251} a^{13} - \frac{323764161}{721445251} a^{8} + \frac{286000}{721445251} a^{7} - \frac{110942160}{721445251} a^{6} + \frac{12870000}{721445251} a^{5} + \frac{103289888}{721445251} a^{4} + \frac{171600000}{721445251} a^{3} + \frac{131343165}{721445251} a^{2} - \frac{149445251}{721445251} a + \frac{235384018}{721445251}$, $\frac{1}{721445251} a^{14} + \frac{364000}{721445251} a^{8} + \frac{170022290}{721445251} a^{7} + \frac{19110000}{721445251} a^{6} - \frac{125119624}{721445251} a^{5} + \frac{327600000}{721445251} a^{4} + \frac{238758952}{721445251} a^{3} - \frac{344335753}{721445251} a^{2} - \frac{163813872}{721445251} a + \frac{117109498}{721445251}$, $\frac{1}{721445251} a^{15} + \frac{128666199}{721445251} a^{8} - \frac{13650000}{721445251} a^{7} + \frac{173619351}{721445251} a^{6} + \frac{66245251}{721445251} a^{5} - \frac{228664434}{721445251} a^{4} + \frac{278788263}{721445251} a^{3} - \frac{295974705}{721445251} a^{2} - \frac{177854207}{721445251} a - \frac{345562354}{721445251}$

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

Class group and class number

$C_{220354}$, which has order $220354$ (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:  \( 3640.01221338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

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

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
41Data not computed