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Properties

Label 16.8.33531062442...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{36}\cdot 5^{12}\cdot 29^{4}\cdot 41^{4}$
Root discriminant $93.40$
Ramified primes $2, 5, 29, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T203)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![447410716, -165073424, -626828664, 180554672, -24759608, -52326304, 33848312, -11880328, 3141706, -425328, 12692, 10836, -3608, 372, -14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 14*x^14 + 372*x^13 - 3608*x^12 + 10836*x^11 + 12692*x^10 - 425328*x^9 + 3141706*x^8 - 11880328*x^7 + 33848312*x^6 - 52326304*x^5 - 24759608*x^4 + 180554672*x^3 - 626828664*x^2 - 165073424*x + 447410716)
 
gp: K = bnfinit(x^16 - 4*x^15 - 14*x^14 + 372*x^13 - 3608*x^12 + 10836*x^11 + 12692*x^10 - 425328*x^9 + 3141706*x^8 - 11880328*x^7 + 33848312*x^6 - 52326304*x^5 - 24759608*x^4 + 180554672*x^3 - 626828664*x^2 - 165073424*x + 447410716, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 14 x^{14} + 372 x^{13} - 3608 x^{12} + 10836 x^{11} + 12692 x^{10} - 425328 x^{9} + 3141706 x^{8} - 11880328 x^{7} + 33848312 x^{6} - 52326304 x^{5} - 24759608 x^{4} + 180554672 x^{3} - 626828664 x^{2} - 165073424 x + 447410716 \)

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:  \(33531062442740678656000000000000=2^{36}\cdot 5^{12}\cdot 29^{4}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.40$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{9} - \frac{2}{5} a^{6} - \frac{2}{5} a^{3} - \frac{2}{5}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{10} - \frac{2}{5} a^{7} - \frac{2}{5} a^{4} - \frac{2}{5} a$, $\frac{1}{1770} a^{14} - \frac{13}{1770} a^{13} + \frac{8}{295} a^{12} + \frac{301}{1770} a^{11} + \frac{247}{1770} a^{10} + \frac{23}{1770} a^{9} + \frac{158}{885} a^{8} + \frac{71}{885} a^{7} + \frac{128}{295} a^{6} + \frac{383}{885} a^{5} - \frac{194}{885} a^{4} + \frac{254}{885} a^{3} - \frac{182}{885} a^{2} - \frac{394}{885} a - \frac{151}{885}$, $\frac{1}{11173688696595245712790217855555982190702287645725075787609870} a^{15} - \frac{108591298007451004161702425946194328599066817013347893314}{5586844348297622856395108927777991095351143822862537893804935} a^{14} + \frac{57203135575058595315332815246372119795863991487397590765001}{2234737739319049142558043571111196438140457529145015157521974} a^{13} - \frac{45038247833583210335563532550806239190057654044175318720366}{1117368869659524571279021785555598219070228764572507578760987} a^{12} - \frac{159462018025222412112150488963823742336273819714082446655118}{1862281449432540952131702975925997031783714607620845964601645} a^{11} + \frac{8377290026802863296953397943834605412502110570264463870619}{37876910835916087162000738493410109121024703883813816229186} a^{10} - \frac{284099858014849858175321506345509864823087565375694635111657}{2234737739319049142558043571111196438140457529145015157521974} a^{9} - \frac{128718356741532479139593402657574259747943364700130072606703}{1862281449432540952131702975925997031783714607620845964601645} a^{8} + \frac{445142617038497695023027892032953772438151065161648679675521}{1117368869659524571279021785555598219070228764572507578760987} a^{7} + \frac{448420517698619050995868525313591342548061373738119123560852}{1117368869659524571279021785555598219070228764572507578760987} a^{6} - \frac{1182232190904517017647498214853839574819058431819120788833374}{5586844348297622856395108927777991095351143822862537893804935} a^{5} - \frac{287002256852147311830603274223113380103486187594644982487289}{1117368869659524571279021785555598219070228764572507578760987} a^{4} + \frac{122607508609171915640566410703937916539452669971895047996667}{1117368869659524571279021785555598219070228764572507578760987} a^{3} + \frac{1254063069680907798276791314112249517067049953129661243463796}{5586844348297622856395108927777991095351143822862537893804935} a^{2} - \frac{97357700090428577841406498525065779623788193867944877273889}{372456289886508190426340595185199406356742921524169192920329} a - \frac{392322722086268532848907826370120620634838801486427057161908}{5586844348297622856395108927777991095351143822862537893804935}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 1635391253.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T203):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 41 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$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$$[\ ]^{4}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
41Data not computed