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Properties

Label 16.0.16285050713...0864.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 7^{8}\cdot 17^{14}$
Root discriminant $89.28$
Ramified primes $2, 7, 17$
Class number $144640$ (GRH)
Class group $[16, 9040]$ (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![3916322369, -435146928, 1594732716, -158389884, 301456646, -26419040, 34526242, -2617358, 2622159, -166672, 135514, -6854, 4673, -170, 99, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 99*x^14 - 170*x^13 + 4673*x^12 - 6854*x^11 + 135514*x^10 - 166672*x^9 + 2622159*x^8 - 2617358*x^7 + 34526242*x^6 - 26419040*x^5 + 301456646*x^4 - 158389884*x^3 + 1594732716*x^2 - 435146928*x + 3916322369)
 
gp: K = bnfinit(x^16 - 2*x^15 + 99*x^14 - 170*x^13 + 4673*x^12 - 6854*x^11 + 135514*x^10 - 166672*x^9 + 2622159*x^8 - 2617358*x^7 + 34526242*x^6 - 26419040*x^5 + 301456646*x^4 - 158389884*x^3 + 1594732716*x^2 - 435146928*x + 3916322369, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 99 x^{14} - 170 x^{13} + 4673 x^{12} - 6854 x^{11} + 135514 x^{10} - 166672 x^{9} + 2622159 x^{8} - 2617358 x^{7} + 34526242 x^{6} - 26419040 x^{5} + 301456646 x^{4} - 158389884 x^{3} + 1594732716 x^{2} - 435146928 x + 3916322369 \)

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:  \(16285050713501206114103002660864=2^{24}\cdot 7^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.28$
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:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(952=2^{3}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{952}(897,·)$, $\chi_{952}(69,·)$, $\chi_{952}(1,·)$, $\chi_{952}(393,·)$, $\chi_{952}(909,·)$, $\chi_{952}(13,·)$, $\chi_{952}(281,·)$, $\chi_{952}(729,·)$, $\chi_{952}(461,·)$, $\chi_{952}(349,·)$, $\chi_{952}(225,·)$, $\chi_{952}(293,·)$, $\chi_{952}(169,·)$, $\chi_{952}(237,·)$, $\chi_{952}(797,·)$, $\chi_{952}(841,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{64597181628488083810940653271054648024601169036769} a^{15} - \frac{3121241320959620447230328838207141919775732596574}{64597181628488083810940653271054648024601169036769} a^{14} + \frac{25886382003262712201500570629567835229925413816799}{64597181628488083810940653271054648024601169036769} a^{13} + \frac{13979196519662591769873660491919011756008215120788}{64597181628488083810940653271054648024601169036769} a^{12} - \frac{17538036610619725551122572420909068710599292381711}{64597181628488083810940653271054648024601169036769} a^{11} + \frac{1846465390180674306846767251193008421078395614576}{64597181628488083810940653271054648024601169036769} a^{10} + \frac{8484674926600408610706548929345970944286263264316}{64597181628488083810940653271054648024601169036769} a^{9} - \frac{15907513655765279480728611226318609513907103632447}{64597181628488083810940653271054648024601169036769} a^{8} + \frac{2764481138165829957140775974395865040234871906212}{64597181628488083810940653271054648024601169036769} a^{7} - \frac{6146980927405663072905371902547717433677570595840}{64597181628488083810940653271054648024601169036769} a^{6} - \frac{4999997639449980701656441558265950994980236613144}{64597181628488083810940653271054648024601169036769} a^{5} + \frac{12339933059506904745439996143635447334770012469701}{64597181628488083810940653271054648024601169036769} a^{4} - \frac{17451586482915275960378991840332411906192307913372}{64597181628488083810940653271054648024601169036769} a^{3} + \frac{10144346536837992463928954796449147630331323313957}{64597181628488083810940653271054648024601169036769} a^{2} - \frac{17793900521677835073861008908903114043303049753}{270281094679866459460002733351693087969042548271} a + \frac{20226359626413805666255303523790897172893567382481}{64597181628488083810940653271054648024601169036769}$

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

Class group and class number

$C_{16}\times C_{9040}$, which has order $144640$ (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:  \( 3640.012213375973 \) (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

\(\Q(\sqrt{-238}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-14}, \sqrt{17})\), 4.4.4913.1, 4.0.15407168.2, 8.0.237380825780224.45, \(\Q(\zeta_{17})^+\), 8.0.4035474038263808.6

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
7Data not computed
17Data not computed