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Properties

Label 16.0.23658405876...801.16
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 67^{8}$
Root discriminant $68.53$
Ramified primes $17, 67$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3143164, 3764514, -1582275, -1944392, 2051118, -473688, -124185, 136034, -30601, -1820, 3527, -686, 51, 54, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 + 54*x^13 + 51*x^12 - 686*x^11 + 3527*x^10 - 1820*x^9 - 30601*x^8 + 136034*x^7 - 124185*x^6 - 473688*x^5 + 2051118*x^4 - 1944392*x^3 - 1582275*x^2 + 3764514*x + 3143164)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 + 54*x^13 + 51*x^12 - 686*x^11 + 3527*x^10 - 1820*x^9 - 30601*x^8 + 136034*x^7 - 124185*x^6 - 473688*x^5 + 2051118*x^4 - 1944392*x^3 - 1582275*x^2 + 3764514*x + 3143164, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 2 x^{14} + 54 x^{13} + 51 x^{12} - 686 x^{11} + 3527 x^{10} - 1820 x^{9} - 30601 x^{8} + 136034 x^{7} - 124185 x^{6} - 473688 x^{5} + 2051118 x^{4} - 1944392 x^{3} - 1582275 x^{2} + 3764514 x + 3143164 \)

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:  \(236584058764743389289008392801=17^{12}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.53$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{16} a^{10} + \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{16} a^{4} + \frac{1}{16} a^{3} + \frac{3}{16} a^{2} - \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{16} a^{11} - \frac{1}{8} a^{9} + \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{16} a^{2} + \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{10336} a^{12} + \frac{101}{5168} a^{11} - \frac{287}{10336} a^{10} + \frac{5}{2584} a^{9} + \frac{17}{608} a^{8} + \frac{313}{2584} a^{7} - \frac{1131}{10336} a^{6} - \frac{47}{2584} a^{5} - \frac{2007}{10336} a^{4} - \frac{75}{304} a^{3} - \frac{79}{608} a^{2} + \frac{49}{304} a + \frac{55}{152}$, $\frac{1}{10336} a^{13} + \frac{253}{10336} a^{11} - \frac{73}{5168} a^{10} + \frac{125}{10336} a^{9} + \frac{507}{5168} a^{8} - \frac{803}{10336} a^{7} - \frac{851}{5168} a^{6} + \frac{2377}{10336} a^{5} - \frac{383}{2584} a^{4} + \frac{49}{608} a^{3} + \frac{5}{152} a^{2} + \frac{1}{19} a + \frac{31}{76}$, $\frac{1}{2087872} a^{14} - \frac{97}{2087872} a^{13} + \frac{47}{1043936} a^{12} + \frac{34927}{2087872} a^{11} - \frac{585}{260984} a^{10} - \frac{94395}{2087872} a^{9} - \frac{595}{30704} a^{8} - \frac{215903}{2087872} a^{7} - \frac{39145}{521968} a^{6} - \frac{156989}{2087872} a^{5} - \frac{194669}{1043936} a^{4} + \frac{34925}{122816} a^{3} - \frac{57063}{122816} a^{2} - \frac{23}{61408} a - \frac{2523}{30704}$, $\frac{1}{15750846049617908264068986046016} a^{15} - \frac{45752416032020738855817}{926520355859876956709940355648} a^{14} + \frac{254679882652863602435986939}{7875423024808954132034493023008} a^{13} + \frac{395330922812500168569942283}{15750846049617908264068986046016} a^{12} - \frac{8845448408482099415786212361}{984427878101119266504311627876} a^{11} + \frac{61135708771018510492628359185}{15750846049617908264068986046016} a^{10} + \frac{69805119842205278292033080775}{3937711512404477066017246511504} a^{9} - \frac{1459526853031526189192476600123}{15750846049617908264068986046016} a^{8} - \frac{86691616565337961455111317359}{3937711512404477066017246511504} a^{7} + \frac{346729691374814328842766584383}{15750846049617908264068986046016} a^{6} + \frac{5376076682800375357859248335}{605801771139150317848807155616} a^{5} + \frac{31240175463010383330968368985}{15750846049617908264068986046016} a^{4} + \frac{8621919981017666030069199817}{926520355859876956709940355648} a^{3} - \frac{145149433045952762599502383225}{463260177929938478354970177824} a^{2} - \frac{369671877090247500331364293}{901284392859802487071926416} a + \frac{1527555258378991260479076909}{57907522241242309794371272228}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 99025228.2163 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-1139}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-67}) \), \(\Q(\sqrt{17}, \sqrt{-67})\), 4.2.19363.1 x2, 4.0.76313.1 x2, 8.0.1683041777041.2, 8.2.7259687665147.2 x4

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$