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Properties

Label 16.0.17624764696...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 29^{6}\cdot 41^{4}$
Root discriminant $119.64$
Ramified primes $2, 5, 29, 41$
Class number $96$ (GRH)
Class group $[2, 2, 2, 12]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T456)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![151994111, 283026660, 225613310, 108407120, 56975359, 16447560, 11172540, 894240, 1710046, -36380, 139950, -4080, 5779, -80, 120, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 120*x^14 - 80*x^13 + 5779*x^12 - 4080*x^11 + 139950*x^10 - 36380*x^9 + 1710046*x^8 + 894240*x^7 + 11172540*x^6 + 16447560*x^5 + 56975359*x^4 + 108407120*x^3 + 225613310*x^2 + 283026660*x + 151994111)
 
gp: K = bnfinit(x^16 + 120*x^14 - 80*x^13 + 5779*x^12 - 4080*x^11 + 139950*x^10 - 36380*x^9 + 1710046*x^8 + 894240*x^7 + 11172540*x^6 + 16447560*x^5 + 56975359*x^4 + 108407120*x^3 + 225613310*x^2 + 283026660*x + 151994111, 1)
 

Normalized defining polynomial

\( x^{16} + 120 x^{14} - 80 x^{13} + 5779 x^{12} - 4080 x^{11} + 139950 x^{10} - 36380 x^{9} + 1710046 x^{8} + 894240 x^{7} + 11172540 x^{6} + 16447560 x^{5} + 56975359 x^{4} + 108407120 x^{3} + 225613310 x^{2} + 283026660 x + 151994111 \)

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:  \(1762476469646556921856000000000000=2^{32}\cdot 5^{12}\cdot 29^{6}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $119.64$
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:  $2, 5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{12} a^{12} - \frac{1}{6} a^{10} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{5}{12}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{12} + \frac{1}{6} a^{10} - \frac{1}{3} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{5}{12} a^{6} + \frac{5}{12} a^{5} - \frac{5}{12} a^{4} + \frac{5}{12} a^{3} + \frac{1}{4} a^{2} - \frac{7}{24} a - \frac{3}{8}$, $\frac{1}{72} a^{14} - \frac{1}{72} a^{13} - \frac{1}{18} a^{11} + \frac{1}{9} a^{10} - \frac{1}{18} a^{9} - \frac{17}{36} a^{8} - \frac{1}{12} a^{7} + \frac{13}{36} a^{6} + \frac{11}{36} a^{5} - \frac{5}{12} a^{4} - \frac{17}{36} a^{3} - \frac{31}{72} a^{2} + \frac{31}{72} a - \frac{2}{9}$, $\frac{1}{326598930092448547117139895078962994347579177338979912208} a^{15} - \frac{781507368397782758527622296372027385658829219676751281}{326598930092448547117139895078962994347579177338979912208} a^{14} - \frac{6380313263690560548016738350791556916253791706680426775}{326598930092448547117139895078962994347579177338979912208} a^{13} - \frac{2815440752314589874006316732913131623443731025279777809}{326598930092448547117139895078962994347579177338979912208} a^{12} - \frac{176598153490477829336209489120828143139781272525581059}{2268048125642003799424582604715020794080410953742916057} a^{11} - \frac{153996626291014677105465668675234705955223235631144023}{13608288753852022796547495628290124764482465722457496342} a^{10} - \frac{20281830549275714550907118642252471046638533291809461195}{54433155015408091186189982513160499057929862889829985368} a^{9} - \frac{26058230324509401579185891419275509379549710334549670897}{163299465046224273558569947539481497173789588669489956104} a^{8} - \frac{19680445013665947378416605837265076603480640084755178457}{40824866261556068389642486884870374293447397167372489026} a^{7} + \frac{1962554546240984768069401250590297742603084787194129432}{20412433130778034194821243442435187146723698583686244513} a^{6} - \frac{24808685822049578616744329012221907403939442718544698725}{81649732523112136779284973769740748586894794334744978052} a^{5} + \frac{12403804453337374808649611287825485927559613527455890525}{81649732523112136779284973769740748586894794334744978052} a^{4} + \frac{1351532370256860523230264067378250149052802572193651793}{12096256670090686930264440558480110901762191753295552304} a^{3} - \frac{9189187890253443014633064177464363393627402897236202131}{326598930092448547117139895078962994347579177338979912208} a^{2} - \frac{107339397300089060580690111792080146612503684415264746335}{326598930092448547117139895078962994347579177338979912208} a + \frac{1708319953405519031377432760246887886367025060449582923}{3669650899915152214799324663808572970197518846505392272}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{12}$, which has order $96$ (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:  \( 167965910.802 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T456):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.3625.1, 4.0.232000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.53824000000.4

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.16$x^{8} + 2 x^{6} + 4 x^{5} + 6 x^{4} + 8 x^{3} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed