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Properties

Label 16.0.82432069367...0721.8
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 29^{12}$
Root discriminant $85.56$
Ramified primes $13, 29$
Class number $72$ (GRH)
Class group $[2, 6, 6]$ (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![849081392, 478872560, 91351640, 159631694, 106741005, 37922114, 13124314, 8212415, 3768300, 378230, -64949, -18754, 820, 369, 10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 10*x^14 + 369*x^13 + 820*x^12 - 18754*x^11 - 64949*x^10 + 378230*x^9 + 3768300*x^8 + 8212415*x^7 + 13124314*x^6 + 37922114*x^5 + 106741005*x^4 + 159631694*x^3 + 91351640*x^2 + 478872560*x + 849081392)
 
gp: K = bnfinit(x^16 - 2*x^15 + 10*x^14 + 369*x^13 + 820*x^12 - 18754*x^11 - 64949*x^10 + 378230*x^9 + 3768300*x^8 + 8212415*x^7 + 13124314*x^6 + 37922114*x^5 + 106741005*x^4 + 159631694*x^3 + 91351640*x^2 + 478872560*x + 849081392, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 10 x^{14} + 369 x^{13} + 820 x^{12} - 18754 x^{11} - 64949 x^{10} + 378230 x^{9} + 3768300 x^{8} + 8212415 x^{7} + 13124314 x^{6} + 37922114 x^{5} + 106741005 x^{4} + 159631694 x^{3} + 91351640 x^{2} + 478872560 x + 849081392 \)

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:  \(8243206936713178643875538610721=13^{12}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.56$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{10} - \frac{1}{3} a^{9} + \frac{1}{6} a^{7} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} + \frac{1}{6} a$, $\frac{1}{10872934832004} a^{14} - \frac{5560165855}{2718233708001} a^{13} - \frac{433841204365}{5436467416002} a^{12} - \frac{4079386347095}{10872934832004} a^{11} - \frac{800601586367}{1812155805334} a^{10} + \frac{70556105635}{1812155805334} a^{9} - \frac{381106380565}{10872934832004} a^{8} - \frac{1142449104364}{2718233708001} a^{7} + \frac{61580551873}{2718233708001} a^{6} - \frac{550049167007}{3624311610668} a^{5} + \frac{112615124825}{388319101143} a^{4} + \frac{384187344479}{776638202286} a^{3} - \frac{207174116215}{517758801524} a^{2} - \frac{238616867416}{2718233708001} a - \frac{983490150929}{2718233708001}$, $\frac{1}{78018359819796247655545747162621026539493590799364945645752} a^{15} + \frac{1099496205830798692252061135326081936578713875}{39009179909898123827772873581310513269746795399682472822876} a^{14} + \frac{2422694137119174715136083447117646880008541349912288793701}{39009179909898123827772873581310513269746795399682472822876} a^{13} + \frac{2955568839611476802839862621773862662911504710484660012083}{26006119939932082551848582387540342179831196933121648548584} a^{12} + \frac{94159401833684834143783639039701432121166235546975963641}{6501529984983020637962145596885085544957799233280412137146} a^{11} + \frac{246682307430877458730307688098685216962047160303837003561}{5572739987128303403967553368758644752820970771383210403268} a^{10} - \frac{7448327151216708391261710329429449661877252064480953660997}{78018359819796247655545747162621026539493590799364945645752} a^{9} + \frac{17847857354357635007357828142122884551272442924205631825687}{39009179909898123827772873581310513269746795399682472822876} a^{8} - \frac{1865338055795838738946514035375194042230868790667222795233}{19504589954949061913886436790655256634873397699841236411438} a^{7} + \frac{26171931987860128800082479153412824167056838369061007473839}{78018359819796247655545747162621026539493590799364945645752} a^{6} - \frac{13353662581727987016595735558612234550300022830940807412007}{39009179909898123827772873581310513269746795399682472822876} a^{5} + \frac{2330865392376436305224557761550759457916152815578344501855}{5572739987128303403967553368758644752820970771383210403268} a^{4} - \frac{738194336177115822783157522287781064976193293212440652653}{11145479974256606807935106737517289505641941542766420806536} a^{3} - \frac{3958893368882962032219895725837613788735778219069443294535}{13003059969966041275924291193770171089915598466560824274292} a^{2} + \frac{4714070570989327691965154908380746678519407188649028075395}{9752294977474530956943218395327628317436698849920618205719} a + \frac{574441627431441299963235502085846975055230477093500447234}{9752294977474530956943218395327628317436698849920618205719}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{6}$, which has order $72$ (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:  \( 8238717.72759 \) (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

\(\Q(\sqrt{377}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{29}) \), 4.4.53582633.1, \(\Q(\sqrt{13}, \sqrt{29})\), 4.4.53582633.2, 4.0.4901.1 x2, 4.0.10933.1 x2, 8.8.2871098559212689.3, 8.0.20200652641.1, 8.0.2871098559212689.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
$29$29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$