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Properties

Label 16.8.30341377703...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{44}\cdot 5^{14}\cdot 41^{4}$
Root discriminant $69.60$
Ramified primes $2, 5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times D_4).C_2^3$ (as 16T315)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2091041, 9102284, 2510564, -15272956, -1022814, 2228892, 579816, 12812, 98082, 41228, -15764, -2012, -714, -244, 24, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 24*x^14 - 244*x^13 - 714*x^12 - 2012*x^11 - 15764*x^10 + 41228*x^9 + 98082*x^8 + 12812*x^7 + 579816*x^6 + 2228892*x^5 - 1022814*x^4 - 15272956*x^3 + 2510564*x^2 + 9102284*x + 2091041)
 
gp: K = bnfinit(x^16 - 4*x^15 + 24*x^14 - 244*x^13 - 714*x^12 - 2012*x^11 - 15764*x^10 + 41228*x^9 + 98082*x^8 + 12812*x^7 + 579816*x^6 + 2228892*x^5 - 1022814*x^4 - 15272956*x^3 + 2510564*x^2 + 9102284*x + 2091041, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 24 x^{14} - 244 x^{13} - 714 x^{12} - 2012 x^{11} - 15764 x^{10} + 41228 x^{9} + 98082 x^{8} + 12812 x^{7} + 579816 x^{6} + 2228892 x^{5} - 1022814 x^{4} - 15272956 x^{3} + 2510564 x^{2} + 9102284 x + 2091041 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(303413777032806400000000000000=2^{44}\cdot 5^{14}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.60$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{66} a^{13} + \frac{5}{66} a^{12} - \frac{7}{33} a^{11} - \frac{4}{33} a^{10} - \frac{7}{66} a^{9} + \frac{3}{22} a^{8} + \frac{3}{11} a^{6} - \frac{1}{6} a^{5} - \frac{23}{66} a^{4} + \frac{1}{33} a^{3} + \frac{5}{11} a^{2} + \frac{19}{66} a - \frac{23}{66}$, $\frac{1}{66} a^{14} + \frac{5}{66} a^{12} + \frac{7}{66} a^{11} - \frac{1}{6} a^{9} - \frac{1}{66} a^{8} - \frac{13}{33} a^{7} + \frac{3}{22} a^{6} - \frac{2}{11} a^{5} + \frac{7}{66} a^{4} + \frac{3}{22} a^{3} - \frac{16}{33} a^{2} - \frac{19}{66} a + \frac{9}{22}$, $\frac{1}{6954065209462880731060491840104889429680755693516926834} a^{15} + \frac{90720163426079358405973481182921233562262445156353}{1159010868243813455176748640017481571613459282252821139} a^{14} + \frac{33580858430820103393224249913642696737829294646820575}{6954065209462880731060491840104889429680755693516926834} a^{13} - \frac{11384258782884319503319745826110852627087738349024869}{632187746314807339187317440009535402698250517592447894} a^{12} + \frac{228950667988240507447078630730330861857331556330922147}{6954065209462880731060491840104889429680755693516926834} a^{11} + \frac{991483980062618454536423294125826785544353104050226553}{6954065209462880731060491840104889429680755693516926834} a^{10} + \frac{196283857238434846573223140721168184612064322263551729}{2318021736487626910353497280034963143226918564505642278} a^{9} - \frac{382122729746205742643346079435354201967307184545703327}{6954065209462880731060491840104889429680755693516926834} a^{8} + \frac{3367141064315952291064822905832576789694928291362601707}{6954065209462880731060491840104889429680755693516926834} a^{7} - \frac{11308006716226965950971419303997264009015894623961690}{28735806650673060872150792727706154668102296254202177} a^{6} + \frac{892966722109059685177885946155756913132945001227019515}{2318021736487626910353497280034963143226918564505642278} a^{5} + \frac{588929773433363708342211992834198195019957156930408119}{6954065209462880731060491840104889429680755693516926834} a^{4} - \frac{150521237411871355071352869966432635379600611101911391}{2318021736487626910353497280034963143226918564505642278} a^{3} + \frac{3429380214694957719459372787210698621889158315861050723}{6954065209462880731060491840104889429680755693516926834} a^{2} - \frac{489582793752974350543683117489956626546261126332344695}{2318021736487626910353497280034963143226918564505642278} a + \frac{2174340976680334749253480118022273051610960409853904423}{6954065209462880731060491840104889429680755693516926834}$

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:  $11$
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:  \( 320896343.571 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4).C_2^3$ (as 16T315):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $(C_2\times D_4).C_2^3$
Character table for $(C_2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), 4.4.8000.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
41Data not computed