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Properties

Label 16.0.96826519964...000.11
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}$
Root discriminant $20.49$
Ramified primes $2, 3, 5, 7$
Class number $4$
Class group $[4]$
Galois group $Q_8 : C_2^2$ (as 16T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![301, -378, 550, 762, 1802, -1716, -618, -252, 1439, -858, 156, 60, -73, 24, 7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 7*x^14 + 24*x^13 - 73*x^12 + 60*x^11 + 156*x^10 - 858*x^9 + 1439*x^8 - 252*x^7 - 618*x^6 - 1716*x^5 + 1802*x^4 + 762*x^3 + 550*x^2 - 378*x + 301)
 
gp: K = bnfinit(x^16 - 6*x^15 + 7*x^14 + 24*x^13 - 73*x^12 + 60*x^11 + 156*x^10 - 858*x^9 + 1439*x^8 - 252*x^7 - 618*x^6 - 1716*x^5 + 1802*x^4 + 762*x^3 + 550*x^2 - 378*x + 301, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 7 x^{14} + 24 x^{13} - 73 x^{12} + 60 x^{11} + 156 x^{10} - 858 x^{9} + 1439 x^{8} - 252 x^{7} - 618 x^{6} - 1716 x^{5} + 1802 x^{4} + 762 x^{3} + 550 x^{2} - 378 x + 301 \)

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:  \(968265199641600000000=2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.49$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{5125329352770783808261533} a^{15} - \frac{37765535951119602529454}{732189907538683401180219} a^{14} - \frac{43877999863413453836162}{732189907538683401180219} a^{13} - \frac{214364712306666163924037}{1708443117590261269420511} a^{12} + \frac{2411589590049121498565620}{5125329352770783808261533} a^{11} + \frac{210948430933596820757530}{465939032070071255296503} a^{10} + \frac{900289543712436331347676}{5125329352770783808261533} a^{9} + \frac{220713795515760543843592}{5125329352770783808261533} a^{8} - \frac{262024140032129535209}{9508959838164719495847} a^{7} + \frac{363472916738249935682029}{732189907538683401180219} a^{6} - \frac{241724306679920649172723}{5125329352770783808261533} a^{5} - \frac{1925239152217873122833089}{5125329352770783808261533} a^{4} - \frac{2163083579870727321499543}{5125329352770783808261533} a^{3} - \frac{237983466220793703566619}{1708443117590261269420511} a^{2} + \frac{184298835051151517805935}{732189907538683401180219} a - \frac{193162336673480853994349}{732189907538683401180219}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( \frac{1270077547752}{519724278891191} a^{15} - \frac{23548318944379}{1559172836673573} a^{14} + \frac{10791063685760}{519724278891191} a^{13} + \frac{26776754487767}{519724278891191} a^{12} - \frac{99621620201874}{519724278891191} a^{11} + \frac{29669863486703}{141742985152143} a^{10} + \frac{164854617231352}{519724278891191} a^{9} - \frac{3419539387182073}{1559172836673573} a^{8} + \frac{196399636185730}{47247661717381} a^{7} - \frac{2949790157808862}{1559172836673573} a^{6} - \frac{649581553565456}{519724278891191} a^{5} - \frac{3516816249816203}{1559172836673573} a^{4} + \frac{1747374602635542}{519724278891191} a^{3} + \frac{468236877079526}{519724278891191} a^{2} + \frac{595403930847754}{519724278891191} a + \frac{740788723595714}{1559172836673573} \) (order $6$)
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:  \( 10360.2378236 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2$ (as 16T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 17 conjugacy class representatives for $Q_8 : C_2^2$
Character table for $Q_8 : C_2^2$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{-3}, \sqrt{35})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-5}, \sqrt{-7})\), \(\Q(\sqrt{15}, \sqrt{21})\), \(\Q(\sqrt{-5}, \sqrt{21})\), \(\Q(\sqrt{-7}, \sqrt{15})\), 8.0.31116960000.3, 8.0.77792400.1 x2, 8.0.635040000.1 x2

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$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}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$