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Properties

Label 16.0.89791815397...0896.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 13^{8}$
Root discriminant $17.66$
Ramified primes $2, 3, 13$
Class number $2$
Class group $[2]$
Galois group $D_4\times C_2$ (as 16T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 18, -88, 261, -146, 50, -10, -108, 76, -24, -2, 53, -28, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 28*x^13 + 53*x^12 - 2*x^11 - 24*x^10 + 76*x^9 - 108*x^8 - 10*x^7 + 50*x^6 - 146*x^5 + 261*x^4 - 88*x^3 + 18*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 28*x^13 + 53*x^12 - 2*x^11 - 24*x^10 + 76*x^9 - 108*x^8 - 10*x^7 + 50*x^6 - 146*x^5 + 261*x^4 - 88*x^3 + 18*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 8 x^{14} - 28 x^{13} + 53 x^{12} - 2 x^{11} - 24 x^{10} + 76 x^{9} - 108 x^{8} - 10 x^{7} + 50 x^{6} - 146 x^{5} + 261 x^{4} - 88 x^{3} + 18 x^{2} - 6 x + 1 \)

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:  \(89791815397090000896=2^{24}\cdot 3^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.66$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{18} a^{13} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{4}{9} a^{8} - \frac{2}{9} a^{6} - \frac{1}{3} a^{5} - \frac{2}{9} a^{4} - \frac{1}{3} a^{3} + \frac{2}{9} a^{2} - \frac{1}{6} a - \frac{2}{9}$, $\frac{1}{413838} a^{14} + \frac{3457}{413838} a^{13} + \frac{27437}{413838} a^{12} + \frac{7478}{206919} a^{11} - \frac{1022}{22991} a^{10} + \frac{78547}{206919} a^{9} + \frac{96209}{206919} a^{8} - \frac{77177}{206919} a^{7} - \frac{20}{747} a^{6} - \frac{62153}{206919} a^{5} + \frac{22204}{206919} a^{4} - \frac{14653}{206919} a^{3} - \frac{111929}{413838} a^{2} - \frac{31345}{413838} a + \frac{183713}{413838}$, $\frac{1}{23767130178} a^{15} + \frac{721}{3961188363} a^{14} - \frac{96846296}{3961188363} a^{13} + \frac{1938590}{516676743} a^{12} - \frac{49995854}{3961188363} a^{11} + \frac{506781079}{3961188363} a^{10} + \frac{5304277910}{11883565089} a^{9} - \frac{1633494509}{3961188363} a^{8} - \frac{705144169}{3961188363} a^{7} + \frac{3446003714}{11883565089} a^{6} + \frac{515716823}{3961188363} a^{5} - \frac{28427000}{172225581} a^{4} + \frac{1094673863}{7922376726} a^{3} + \frac{888150038}{3961188363} a^{2} + \frac{1443280070}{3961188363} a - \frac{3445329254}{11883565089}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{4957783102}{11883565089} a^{15} + \frac{12636334541}{7922376726} a^{14} - \frac{36188101511}{11883565089} a^{13} + \frac{5735328002}{516676743} a^{12} - \frac{238439687029}{11883565089} a^{11} - \frac{35704621015}{11883565089} a^{10} + \frac{39857355037}{3961188363} a^{9} - \frac{356044289906}{11883565089} a^{8} + \frac{156446339762}{3961188363} a^{7} + \frac{138249837940}{11883565089} a^{6} - \frac{79132239821}{3961188363} a^{5} + \frac{29600926387}{516676743} a^{4} - \frac{129779108649}{1320396121} a^{3} + \frac{439208418887}{23767130178} a^{2} - \frac{5497296557}{3961188363} a + \frac{4935594205}{3961188363} \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 14943.0195941 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4$ (as 16T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{39}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(i, \sqrt{39})\), \(\Q(i, \sqrt{13})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{3}, \sqrt{-13})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{-13})\), \(\Q(\sqrt{3}, \sqrt{13})\), 4.0.8112.1 x2, 4.4.8112.1 x2, 4.4.7488.1 x2, 4.0.7488.1 x2, 8.0.592240896.1, 8.0.1052872704.3 x2, 8.0.56070144.2 x2, 8.0.9475854336.6 x2, 8.0.592240896.6 x2, 8.8.9475854336.1, 8.0.9475854336.3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$