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Properties

Label 16.16.5054886175...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{40}\cdot 3^{12}\cdot 5^{4}\cdot 7^{12}$
Root discriminant $82.98$
Ramified primes $2, 3, 5, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2 \times Q_8):C_2$ (as 16T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16857, 305568, 1229976, 1605240, -380520, -2383248, -1289736, 453156, 446455, -9328, -56336, -3220, 3304, 224, -92, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 92*x^14 + 224*x^13 + 3304*x^12 - 3220*x^11 - 56336*x^10 - 9328*x^9 + 446455*x^8 + 453156*x^7 - 1289736*x^6 - 2383248*x^5 - 380520*x^4 + 1605240*x^3 + 1229976*x^2 + 305568*x + 16857)
 
gp: K = bnfinit(x^16 - 4*x^15 - 92*x^14 + 224*x^13 + 3304*x^12 - 3220*x^11 - 56336*x^10 - 9328*x^9 + 446455*x^8 + 453156*x^7 - 1289736*x^6 - 2383248*x^5 - 380520*x^4 + 1605240*x^3 + 1229976*x^2 + 305568*x + 16857, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 92 x^{14} + 224 x^{13} + 3304 x^{12} - 3220 x^{11} - 56336 x^{10} - 9328 x^{9} + 446455 x^{8} + 453156 x^{7} - 1289736 x^{6} - 2383248 x^{5} - 380520 x^{4} + 1605240 x^{3} + 1229976 x^{2} + 305568 x + 16857 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5054886175427630513006837760000=2^{40}\cdot 3^{12}\cdot 5^{4}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.98$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{21} a^{8} - \frac{2}{21} a^{7} - \frac{2}{7} a^{6} - \frac{5}{21} a^{5} + \frac{1}{3} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{21} a^{9} - \frac{10}{21} a^{7} + \frac{4}{21} a^{6} - \frac{1}{7} a^{5} + \frac{5}{21} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{21} a^{10} + \frac{5}{21} a^{7} - \frac{1}{7} a^{5} + \frac{4}{21} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{21} a^{11} + \frac{10}{21} a^{7} + \frac{2}{7} a^{6} + \frac{8}{21} a^{5} - \frac{2}{21} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7}$, $\frac{1}{21} a^{12} + \frac{5}{21} a^{7} + \frac{5}{21} a^{6} + \frac{2}{7} a^{5} - \frac{10}{21} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{273} a^{13} - \frac{2}{91} a^{12} - \frac{5}{273} a^{11} + \frac{1}{91} a^{10} - \frac{1}{91} a^{9} - \frac{2}{273} a^{8} + \frac{2}{21} a^{7} + \frac{6}{91} a^{6} - \frac{38}{91} a^{5} + \frac{37}{91} a^{4} - \frac{1}{7} a^{3} - \frac{8}{91} a^{2} - \frac{18}{91} a + \frac{34}{91}$, $\frac{1}{273} a^{14} - \frac{2}{273} a^{12} - \frac{1}{273} a^{11} + \frac{2}{273} a^{10} + \frac{2}{91} a^{9} + \frac{1}{273} a^{8} + \frac{19}{91} a^{7} - \frac{19}{273} a^{6} - \frac{5}{13} a^{5} - \frac{101}{273} a^{4} - \frac{34}{91} a^{3} + \frac{38}{91} a^{2} - \frac{5}{13} a - \frac{30}{91}$, $\frac{1}{169959834339212939683079942979} a^{15} + \frac{268836529352243663836096417}{169959834339212939683079942979} a^{14} - \frac{171268946312853524015063734}{169959834339212939683079942979} a^{13} + \frac{451555389290256238159034417}{56653278113070979894359980993} a^{12} - \frac{503163806381052880216772077}{169959834339212939683079942979} a^{11} + \frac{3569877047636139618044542}{13073833410708687667929226383} a^{10} + \frac{793658643569088311860950288}{56653278113070979894359980993} a^{9} + \frac{296366286968409060961557430}{56653278113070979894359980993} a^{8} - \frac{61196154625678929617566285667}{169959834339212939683079942979} a^{7} + \frac{53005517790131347079746548559}{169959834339212939683079942979} a^{6} - \frac{1584086043725670782472301820}{3616166688068360418788934957} a^{5} - \frac{4363227545154761388116311672}{169959834339212939683079942979} a^{4} - \frac{1377724128171756735892284027}{4357944470236229222643075461} a^{3} - \frac{17804674098427026771933037201}{56653278113070979894359980993} a^{2} + \frac{26665417861767058562052253068}{56653278113070979894359980993} a + \frac{2239119352299741085983420604}{8093325444724425699194282999}$

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:  $15$
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:  \( 8268264991.76 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^3$ (as 16T20):

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 $(C_2 \times Q_8):C_2$
Character table for $(C_2 \times Q_8):C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{6}, \sqrt{14})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{14})\), 8.8.12745506816.1

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 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 ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$7$7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$