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Properties

Label 16.8.31236842166...5728.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 17^{15}\cdot 6529^{2}$
Root discriminant $60.38$
Ramified primes $2, 17, 6529$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1223

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4029307, 8481940, -4525330, -2529391, -130335, 1031825, -166320, -166563, 65772, -4635, -325, 1270, -514, 58, -8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 8*x^14 + 58*x^13 - 514*x^12 + 1270*x^11 - 325*x^10 - 4635*x^9 + 65772*x^8 - 166563*x^7 - 166320*x^6 + 1031825*x^5 - 130335*x^4 - 2529391*x^3 - 4525330*x^2 + 8481940*x + 4029307)
 
gp: K = bnfinit(x^16 - 3*x^15 - 8*x^14 + 58*x^13 - 514*x^12 + 1270*x^11 - 325*x^10 - 4635*x^9 + 65772*x^8 - 166563*x^7 - 166320*x^6 + 1031825*x^5 - 130335*x^4 - 2529391*x^3 - 4525330*x^2 + 8481940*x + 4029307, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 8 x^{14} + 58 x^{13} - 514 x^{12} + 1270 x^{11} - 325 x^{10} - 4635 x^{9} + 65772 x^{8} - 166563 x^{7} - 166320 x^{6} + 1031825 x^{5} - 130335 x^{4} - 2529391 x^{3} - 4525330 x^{2} + 8481940 x + 4029307 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31236842166910780816202985728=2^{8}\cdot 17^{15}\cdot 6529^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.38$
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:  $2, 17, 6529$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1776549692290727147127245049840981568960635965273617321} a^{15} - \frac{727174006400586670577303513895250267545617580829014172}{1776549692290727147127245049840981568960635965273617321} a^{14} - \frac{604786865463823204122836443269562762731791041119544629}{1776549692290727147127245049840981568960635965273617321} a^{13} - \frac{865158660077188743531505010778252437570529170593680175}{1776549692290727147127245049840981568960635965273617321} a^{12} + \frac{808078500996631842786687169496951221102117369837477985}{1776549692290727147127245049840981568960635965273617321} a^{11} + \frac{26526298254576695244088126158912410148370976221306274}{1776549692290727147127245049840981568960635965273617321} a^{10} - \frac{673632660400109922682394355296624883319704282976588222}{1776549692290727147127245049840981568960635965273617321} a^{9} + \frac{489448832732417975341633616033183035387775468711618550}{1776549692290727147127245049840981568960635965273617321} a^{8} + \frac{320726247801863844481238339110798390110704985317863147}{1776549692290727147127245049840981568960635965273617321} a^{7} + \frac{1555783349174613222304035719220688776410460914155383}{26515667049115330554137985818522112969561730824979363} a^{6} + \frac{433607850340444801686182334040529053369453951807177676}{1776549692290727147127245049840981568960635965273617321} a^{5} + \frac{273931494190036040799325468062159195105685772196691709}{1776549692290727147127245049840981568960635965273617321} a^{4} - \frac{419154686802151092111766013945610936890641324065776147}{1776549692290727147127245049840981568960635965273617321} a^{3} + \frac{786868023496493802315216541438280681727823034084230522}{1776549692290727147127245049840981568960635965273617321} a^{2} + \frac{805903204928299576064937823816036409992341951081488190}{1776549692290727147127245049840981568960635965273617321} a - \frac{503550039407688189018257505542593742877830916884559934}{1776549692290727147127245049840981568960635965273617321}$

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:  $11$
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:  \( 80374572.4423 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1223:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1223
Character table for t16n1223 is not computed

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.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: 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 R $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
17Data not computed
6529Data not computed