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Properties

Label 16.0.61841248347...401.15
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 61^{12}$
Root discriminant $149.44$
Ramified primes $13, 61$
Class number $512$ (GRH)
Class group $[2, 2, 4, 4, 8]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15614639, 24448631, 5579973, -9463368, 6144281, -126984, 8460029, -3186879, 2534757, -437329, 227730, -23274, 8811, -518, 153, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 153*x^14 - 518*x^13 + 8811*x^12 - 23274*x^11 + 227730*x^10 - 437329*x^9 + 2534757*x^8 - 3186879*x^7 + 8460029*x^6 - 126984*x^5 + 6144281*x^4 - 9463368*x^3 + 5579973*x^2 + 24448631*x + 15614639)
 
gp: K = bnfinit(x^16 - 4*x^15 + 153*x^14 - 518*x^13 + 8811*x^12 - 23274*x^11 + 227730*x^10 - 437329*x^9 + 2534757*x^8 - 3186879*x^7 + 8460029*x^6 - 126984*x^5 + 6144281*x^4 - 9463368*x^3 + 5579973*x^2 + 24448631*x + 15614639, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 153 x^{14} - 518 x^{13} + 8811 x^{12} - 23274 x^{11} + 227730 x^{10} - 437329 x^{9} + 2534757 x^{8} - 3186879 x^{7} + 8460029 x^{6} - 126984 x^{5} + 6144281 x^{4} - 9463368 x^{3} + 5579973 x^{2} + 24448631 x + 15614639 \)

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:  \(61841248347955294332328765542314401=13^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $149.44$
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:  $13, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{36} a^{8} - \frac{1}{18} a^{7} - \frac{1}{9} a^{5} + \frac{5}{36} a^{4} + \frac{5}{18} a^{3} + \frac{17}{36} a^{2} + \frac{1}{12} a + \frac{11}{36}$, $\frac{1}{36} a^{9} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{12} a^{5} - \frac{4}{9} a^{4} + \frac{1}{36} a^{3} + \frac{1}{36} a^{2} + \frac{17}{36} a - \frac{7}{18}$, $\frac{1}{36} a^{10} - \frac{1}{12} a^{6} + \frac{1}{9} a^{5} + \frac{1}{4} a^{4} + \frac{5}{36} a^{3} - \frac{11}{36} a^{2} + \frac{5}{18} a - \frac{1}{9}$, $\frac{1}{972} a^{11} + \frac{2}{243} a^{10} - \frac{7}{972} a^{9} - \frac{2}{243} a^{8} - \frac{151}{972} a^{7} + \frac{20}{243} a^{6} + \frac{11}{486} a^{5} + \frac{413}{972} a^{4} - \frac{137}{486} a^{3} + \frac{55}{972} a^{2} - \frac{115}{972} a - \frac{23}{486}$, $\frac{1}{797040} a^{12} - \frac{1}{265680} a^{11} - \frac{563}{79704} a^{10} + \frac{293}{26568} a^{9} - \frac{37}{8856} a^{8} + \frac{89707}{797040} a^{7} + \frac{15869}{265680} a^{6} - \frac{1171}{11070} a^{5} - \frac{2819}{797040} a^{4} + \frac{15341}{88560} a^{3} - \frac{28433}{88560} a^{2} + \frac{22441}{797040} a + \frac{123167}{797040}$, $\frac{1}{797040} a^{13} + \frac{101}{797040} a^{11} - \frac{323}{39852} a^{10} + \frac{125}{19926} a^{9} - \frac{10483}{797040} a^{8} - \frac{4663}{199260} a^{7} + \frac{30629}{797040} a^{6} + \frac{13957}{159408} a^{5} - \frac{8501}{99630} a^{4} - \frac{19675}{79704} a^{3} - \frac{14173}{79704} a^{2} - \frac{13751}{79704} a + \frac{282581}{797040}$, $\frac{1}{2391120} a^{14} - \frac{1}{2391120} a^{12} - \frac{23}{132840} a^{11} + \frac{263}{119556} a^{10} + \frac{26917}{2391120} a^{9} - \frac{1061}{199260} a^{8} + \frac{48491}{478224} a^{7} + \frac{278191}{2391120} a^{6} + \frac{3257}{49815} a^{5} + \frac{277717}{597780} a^{4} + \frac{44932}{149445} a^{3} + \frac{27613}{298890} a^{2} - \frac{76121}{2391120} a - \frac{291827}{1195560}$, $\frac{1}{6289480472494443919385349658264560} a^{15} - \frac{46597448414821643751233597}{1572370118123610979846337414566140} a^{14} + \frac{657356152234516564133057243}{6289480472494443919385349658264560} a^{13} - \frac{120088037694890947519212281}{1257896094498888783877069931652912} a^{12} + \frac{63403033130405018822913332833}{6289480472494443919385349658264560} a^{11} + \frac{1812694490588742706042229466407}{6289480472494443919385349658264560} a^{10} - \frac{42563797292717200446314109096979}{3144740236247221959692674829132280} a^{9} - \frac{41594526882662266581116986969691}{6289480472494443919385349658264560} a^{8} - \frac{81760011921540718969761226553035}{628948047249444391938534965826456} a^{7} - \frac{161429385661130479261963842227309}{6289480472494443919385349658264560} a^{6} - \frac{45815177964205885115524134968273}{1572370118123610979846337414566140} a^{5} - \frac{2904712481392726259428790552598763}{6289480472494443919385349658264560} a^{4} + \frac{12346342377517836510519763972919}{2096493490831481306461783219421520} a^{3} - \frac{24798677897632971350509506442999}{69883116361049376882059440647384} a^{2} + \frac{34615662281983276689993844881367}{139766232722098753764118881294768} a + \frac{1424496376566230920385205673928003}{6289480472494443919385349658264560}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{8}$, which has order $512$ (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:  \( 308589653.463 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{793}) \), \(\Q(\sqrt{13}) \), 4.0.38359789.2, \(\Q(\sqrt{13}, \sqrt{61})\), 4.0.226981.1, 4.0.8175037.1 x2, 4.0.134017.1 x2, 8.0.1471473412124521.5, 8.0.66831229951369.6, 8.8.248679006649044049.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$61$61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$