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Properties

Label 16.0.58432555897...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{12}\cdot 5^{8}$
Root discriminant $40.78$
Ramified primes $2, 3, 5$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $(C_2^2\times C_4):C_2$ (as 16T54)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4201, 5320, -5852, 13664, 7088, -32216, 53360, -54352, 41656, -24328, 11516, -4232, 1316, -304, 64, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 64*x^14 - 304*x^13 + 1316*x^12 - 4232*x^11 + 11516*x^10 - 24328*x^9 + 41656*x^8 - 54352*x^7 + 53360*x^6 - 32216*x^5 + 7088*x^4 + 13664*x^3 - 5852*x^2 + 5320*x + 4201)
 
gp: K = bnfinit(x^16 - 8*x^15 + 64*x^14 - 304*x^13 + 1316*x^12 - 4232*x^11 + 11516*x^10 - 24328*x^9 + 41656*x^8 - 54352*x^7 + 53360*x^6 - 32216*x^5 + 7088*x^4 + 13664*x^3 - 5852*x^2 + 5320*x + 4201, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 64 x^{14} - 304 x^{13} + 1316 x^{12} - 4232 x^{11} + 11516 x^{10} - 24328 x^{9} + 41656 x^{8} - 54352 x^{7} + 53360 x^{6} - 32216 x^{5} + 7088 x^{4} + 13664 x^{3} - 5852 x^{2} + 5320 x + 4201 \)

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:  \(58432555897690521600000000=2^{48}\cdot 3^{12}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.78$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} - \frac{4}{23} a^{13} + \frac{3}{23} a^{11} - \frac{6}{23} a^{10} - \frac{7}{23} a^{9} - \frac{3}{23} a^{7} - \frac{9}{23} a^{6} - \frac{10}{23} a^{5} + \frac{1}{23} a^{4} + \frac{8}{23} a^{3} + \frac{11}{23} a^{2} + \frac{7}{23} a - \frac{4}{23}$, $\frac{1}{62457458684098527924323638549657} a^{15} + \frac{326795252736978432282953800084}{62457458684098527924323638549657} a^{14} + \frac{24560991610655052255149670285812}{62457458684098527924323638549657} a^{13} + \frac{8395609396155262616217424149516}{62457458684098527924323638549657} a^{12} + \frac{9202018408749136567609698006223}{62457458684098527924323638549657} a^{11} - \frac{10518258636189457733058491260601}{62457458684098527924323638549657} a^{10} + \frac{18689919324603854914787237297934}{62457458684098527924323638549657} a^{9} - \frac{24164018119275086589002309385055}{62457458684098527924323638549657} a^{8} + \frac{28419034404553818812298823918242}{62457458684098527924323638549657} a^{7} - \frac{519601163151038459024450342012}{62457458684098527924323638549657} a^{6} + \frac{321086438724792039115065847217}{62457458684098527924323638549657} a^{5} - \frac{6344098396384531564714573363307}{62457458684098527924323638549657} a^{4} - \frac{483293464720937196500261526350}{2715541681917327301057549502159} a^{3} + \frac{275123274102913818725527326589}{62457458684098527924323638549657} a^{2} - \frac{10953930932862900165175726735015}{62457458684098527924323638549657} a + \frac{11200856641190420600609882837759}{62457458684098527924323638549657}$

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:  \( 178556.167066 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4):C_2$ (as 16T54):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $(C_2^2\times C_4):C_2$
Character table for $(C_2^2\times C_4):C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.4.691200.2 x2, \(\Q(\sqrt{2}, \sqrt{3})\), 4.4.345600.1 x2, 8.8.1911029760000.6, 8.0.19110297600.3, 8.0.8493465600.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
$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}$