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Properties

Label 16.0.29476538281...9968.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 193^{5}$
Root discriminant $25.37$
Ramified primes $2, 3, 193$
Class number $2$
Class group $[2]$
Galois group 16T1163

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![67369, -45732, 71058, -21790, 13740, 12316, -7330, 9272, -1818, 1102, 756, -314, 299, -76, 32, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 32*x^14 - 76*x^13 + 299*x^12 - 314*x^11 + 756*x^10 + 1102*x^9 - 1818*x^8 + 9272*x^7 - 7330*x^6 + 12316*x^5 + 13740*x^4 - 21790*x^3 + 71058*x^2 - 45732*x + 67369)
 
gp: K = bnfinit(x^16 - 4*x^15 + 32*x^14 - 76*x^13 + 299*x^12 - 314*x^11 + 756*x^10 + 1102*x^9 - 1818*x^8 + 9272*x^7 - 7330*x^6 + 12316*x^5 + 13740*x^4 - 21790*x^3 + 71058*x^2 - 45732*x + 67369, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 32 x^{14} - 76 x^{13} + 299 x^{12} - 314 x^{11} + 756 x^{10} + 1102 x^{9} - 1818 x^{8} + 9272 x^{7} - 7330 x^{6} + 12316 x^{5} + 13740 x^{4} - 21790 x^{3} + 71058 x^{2} - 45732 x + 67369 \)

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:  \(29476538281722504019968=2^{24}\cdot 3^{8}\cdot 193^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.37$
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:  $2, 3, 193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{111} a^{14} + \frac{4}{111} a^{13} + \frac{13}{111} a^{12} + \frac{3}{37} a^{11} + \frac{4}{111} a^{10} + \frac{12}{37} a^{9} + \frac{26}{111} a^{8} + \frac{29}{111} a^{7} - \frac{26}{111} a^{6} + \frac{1}{3} a^{5} - \frac{47}{111} a^{4} + \frac{26}{111} a^{3} + \frac{28}{111} a^{2} + \frac{11}{111} a - \frac{9}{37}$, $\frac{1}{80658731321089264789008723} a^{15} - \frac{265271717282622170971595}{80658731321089264789008723} a^{14} + \frac{6451744743914450433665998}{80658731321089264789008723} a^{13} - \frac{3497014028641589082664478}{80658731321089264789008723} a^{12} + \frac{12619038608523927031706761}{80658731321089264789008723} a^{11} + \frac{1798377161295014625756299}{26886243773696421596336241} a^{10} - \frac{1590834814709613366097013}{8962081257898807198778747} a^{9} - \frac{36070285341596290364378624}{80658731321089264789008723} a^{8} - \frac{32547991846493091995492734}{80658731321089264789008723} a^{7} - \frac{1546495147387741267114676}{26886243773696421596336241} a^{6} + \frac{20140182598304383575670337}{80658731321089264789008723} a^{5} + \frac{7041973592528851128214808}{80658731321089264789008723} a^{4} - \frac{39687408512066462911124294}{80658731321089264789008723} a^{3} + \frac{37612504743695013132928969}{80658731321089264789008723} a^{2} - \frac{9190423488855345432583645}{80658731321089264789008723} a - \frac{19530275426988916587688205}{80658731321089264789008723}$

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:  \( \frac{16339542783272876230}{25356407205623786478783} a^{15} - \frac{69445883886377538737}{25356407205623786478783} a^{14} + \frac{481466756566980964732}{25356407205623786478783} a^{13} - \frac{1156285645769685818543}{25356407205623786478783} a^{12} + \frac{3608685837558544280077}{25356407205623786478783} a^{11} - \frac{340533963283089938120}{2817378578402642942087} a^{10} + \frac{862361286324718758253}{8452135735207928826261} a^{9} + \frac{20409911935892280643987}{25356407205623786478783} a^{8} - \frac{40883535222974959928428}{25356407205623786478783} a^{7} + \frac{28475332892491786830971}{8452135735207928826261} a^{6} - \frac{50788310224162145388430}{25356407205623786478783} a^{5} - \frac{18088791634169733856066}{25356407205623786478783} a^{4} + \frac{241811562761300453525623}{25356407205623786478783} a^{3} - \frac{344411313485825857044767}{25356407205623786478783} a^{2} + \frac{457625962924181610450752}{25356407205623786478783} a - \frac{235380645522964534775069}{25356407205623786478783} \) (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:  \( 125207.3212 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1163:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 52 conjugacy class representatives for t16n1163 are not computed
Character table for t16n1163 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{12})\), 8.0.4002048.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$3.4.2.1$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}$
193Data not computed