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Properties

Label 16.4.10787325443...625.29
Degree $16$
Signature $[4, 6]$
Discriminant $5^{10}\cdot 101^{10}$
Root discriminant $48.93$
Ramified primes $5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4451, 13682, -16047, -31738, 96324, -100628, 45782, -1792, -11262, 7928, -4833, 1476, -345, 26, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 26*x^13 - 345*x^12 + 1476*x^11 - 4833*x^10 + 7928*x^9 - 11262*x^8 - 1792*x^7 + 45782*x^6 - 100628*x^5 + 96324*x^4 - 31738*x^3 - 16047*x^2 + 13682*x + 4451)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 + 26*x^13 - 345*x^12 + 1476*x^11 - 4833*x^10 + 7928*x^9 - 11262*x^8 - 1792*x^7 + 45782*x^6 - 100628*x^5 + 96324*x^4 - 31738*x^3 - 16047*x^2 + 13682*x + 4451, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} + 26 x^{13} - 345 x^{12} + 1476 x^{11} - 4833 x^{10} + 7928 x^{9} - 11262 x^{8} - 1792 x^{7} + 45782 x^{6} - 100628 x^{5} + 96324 x^{4} - 31738 x^{3} - 16047 x^{2} + 13682 x + 4451 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1078732544346879404306640625=5^{10}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.93$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{1}{2} a^{6} + \frac{3}{10} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{2} - \frac{1}{10} a - \frac{1}{10}$, $\frac{1}{10} a^{9} + \frac{1}{10} a^{7} + \frac{3}{10} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{3}{10} a^{2} - \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{7} - \frac{3}{10} a^{6} - \frac{1}{10} a^{5} - \frac{1}{5} a^{4} - \frac{3}{10} a^{3} + \frac{3}{10} a^{2} - \frac{1}{10} a + \frac{1}{10}$, $\frac{1}{10} a^{11} - \frac{3}{10} a^{7} + \frac{2}{5} a^{6} + \frac{3}{10} a^{5} - \frac{3}{10} a^{4} + \frac{3}{10} a^{3} - \frac{1}{10} a^{2} - \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{150} a^{12} - \frac{1}{50} a^{11} - \frac{1}{25} a^{10} - \frac{1}{30} a^{9} - \frac{7}{150} a^{8} - \frac{37}{75} a^{7} + \frac{2}{75} a^{6} + \frac{26}{75} a^{5} - \frac{2}{15} a^{4} + \frac{4}{75} a^{3} + \frac{2}{5} a^{2} - \frac{14}{75} a + \frac{43}{150}$, $\frac{1}{150} a^{13} + \frac{7}{150} a^{10} - \frac{7}{150} a^{9} - \frac{1}{30} a^{8} + \frac{11}{75} a^{7} - \frac{71}{150} a^{6} + \frac{1}{150} a^{5} - \frac{37}{150} a^{4} - \frac{17}{50} a^{3} - \frac{29}{75} a^{2} + \frac{17}{75} a - \frac{6}{25}$, $\frac{1}{26850} a^{14} - \frac{13}{4475} a^{13} + \frac{37}{26850} a^{12} - \frac{959}{26850} a^{11} - \frac{67}{2685} a^{10} + \frac{283}{13425} a^{9} + \frac{11}{8950} a^{8} - \frac{2117}{5370} a^{7} - \frac{5909}{13425} a^{6} - \frac{1067}{8950} a^{5} - \frac{253}{537} a^{4} - \frac{1687}{13425} a^{3} + \frac{2894}{13425} a^{2} + \frac{10781}{26850} a + \frac{3529}{26850}$, $\frac{1}{1655479530651138790243031850} a^{15} - \frac{500572564314330402189}{275913255108523131707171975} a^{14} + \frac{292469499407408809649998}{275913255108523131707171975} a^{13} + \frac{170293543035497929511081}{275913255108523131707171975} a^{12} - \frac{63239881472022706953149467}{1655479530651138790243031850} a^{11} - \frac{11288417835527939467205431}{331095906130227758048606370} a^{10} - \frac{8503502588143572800855989}{275913255108523131707171975} a^{9} - \frac{26321345822640082242705637}{1655479530651138790243031850} a^{8} - \frac{51497098381692066554842737}{110365302043409252682868790} a^{7} - \frac{158367019241443766010481223}{1655479530651138790243031850} a^{6} - \frac{13501342785849316006818577}{1655479530651138790243031850} a^{5} - \frac{3038961513679082128388309}{66219181226045551609721274} a^{4} + \frac{20338550050923286939476755}{66219181226045551609721274} a^{3} - \frac{79577747675855571852421811}{275913255108523131707171975} a^{2} + \frac{238469328571266138155104172}{827739765325569395121515925} a - \frac{722266101963224775405394021}{1655479530651138790243031850}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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:  \( 54131543.2062 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.4.65037750625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
101Data not computed