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Properties

Label 16.0.48736115900...5625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 41^{8}$
Root discriminant $26.18$
Ramified primes $5, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T158)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5555, -25615, 56890, -58345, 19365, 12050, -8180, -7000, 10281, -4847, 458, 616, -325, 51, 13, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 13*x^14 + 51*x^13 - 325*x^12 + 616*x^11 + 458*x^10 - 4847*x^9 + 10281*x^8 - 7000*x^7 - 8180*x^6 + 12050*x^5 + 19365*x^4 - 58345*x^3 + 56890*x^2 - 25615*x + 5555)
 
gp: K = bnfinit(x^16 - 7*x^15 + 13*x^14 + 51*x^13 - 325*x^12 + 616*x^11 + 458*x^10 - 4847*x^9 + 10281*x^8 - 7000*x^7 - 8180*x^6 + 12050*x^5 + 19365*x^4 - 58345*x^3 + 56890*x^2 - 25615*x + 5555, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 13 x^{14} + 51 x^{13} - 325 x^{12} + 616 x^{11} + 458 x^{10} - 4847 x^{9} + 10281 x^{8} - 7000 x^{7} - 8180 x^{6} + 12050 x^{5} + 19365 x^{4} - 58345 x^{3} + 56890 x^{2} - 25615 x + 5555 \)

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:  \(48736115900396728515625=5^{14}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.18$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{341} a^{14} - \frac{74}{341} a^{13} + \frac{10}{31} a^{12} + \frac{142}{341} a^{11} + \frac{28}{341} a^{10} + \frac{26}{341} a^{9} + \frac{1}{11} a^{8} + \frac{21}{341} a^{7} + \frac{39}{341} a^{6} + \frac{14}{31} a^{5} - \frac{67}{341} a^{4} + \frac{72}{341} a^{3} - \frac{90}{341} a^{2} + \frac{73}{341} a + \frac{14}{31}$, $\frac{1}{4037854871152135722327353011} a^{15} - \frac{1359372353217401655508677}{4037854871152135722327353011} a^{14} + \frac{1707185227822634332103956032}{4037854871152135722327353011} a^{13} - \frac{346441654009902953968722389}{4037854871152135722327353011} a^{12} - \frac{194526872407380338616879493}{4037854871152135722327353011} a^{11} + \frac{22094488544429876441624822}{212518677429059774859334369} a^{10} + \frac{1882637262085276164544777264}{4037854871152135722327353011} a^{9} - \frac{1398217920530244714903504742}{4037854871152135722327353011} a^{8} + \frac{1392168840127088417577190020}{4037854871152135722327353011} a^{7} - \frac{1162753792451353319780973771}{4037854871152135722327353011} a^{6} + \frac{1897244528502786555248832180}{4037854871152135722327353011} a^{5} + \frac{1357835531342022852032485849}{4037854871152135722327353011} a^{4} + \frac{15527580996457113167542515}{33370701414480460515102091} a^{3} - \frac{5568083823366278299954667}{212518677429059774859334369} a^{2} + \frac{1178890333062532606830929610}{4037854871152135722327353011} a - \frac{78490828536105236072685376}{367077715559285065666123001}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{85800058747334988590717}{130253382940391474913785581} a^{15} - \frac{478467563926915695221128}{130253382940391474913785581} a^{14} + \frac{376547225127083671748611}{130253382940391474913785581} a^{13} + \frac{5215078424206815295915315}{130253382940391474913785581} a^{12} - \frac{20570938158129908603406745}{130253382940391474913785581} a^{11} + \frac{1050010970856532882393281}{6855441207389024995462399} a^{10} + \frac{80051502663734380368207705}{130253382940391474913785581} a^{9} - \frac{309491583960934896420476860}{130253382940391474913785581} a^{8} + \frac{383470350408139128305281223}{130253382940391474913785581} a^{7} + \frac{128515587332448549742942150}{130253382940391474913785581} a^{6} - \frac{693223885192823854923068803}{130253382940391474913785581} a^{5} - \frac{117165169500258112863822926}{130253382940391474913785581} a^{4} + \frac{171289367989039534031947496}{11841216630944679537616871} a^{3} - \frac{110924548250287532802897839}{6855441207389024995462399} a^{2} + \frac{799306587725417614089746035}{130253382940391474913785581} a - \frac{3930390127566433302308586}{11841216630944679537616871} \) (order $10$)
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:  \( 120397.53998 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T158):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.5125.1, 4.0.1025.1, 8.0.5384453125.1 x2, 8.4.220762578125.1 x2, 8.0.26265625.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ 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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$