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Properties

Label 16.0.11654783511...6241.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 29^{8}$
Root discriminant $36.87$
Ramified primes $13, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2304, -8640, 19216, -15244, 21841, -7705, -8494, 12374, -943, -2880, 527, 139, 29, 7, -11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 11*x^14 + 7*x^13 + 29*x^12 + 139*x^11 + 527*x^10 - 2880*x^9 - 943*x^8 + 12374*x^7 - 8494*x^6 - 7705*x^5 + 21841*x^4 - 15244*x^3 + 19216*x^2 - 8640*x + 2304)
 
gp: K = bnfinit(x^16 - 2*x^15 - 11*x^14 + 7*x^13 + 29*x^12 + 139*x^11 + 527*x^10 - 2880*x^9 - 943*x^8 + 12374*x^7 - 8494*x^6 - 7705*x^5 + 21841*x^4 - 15244*x^3 + 19216*x^2 - 8640*x + 2304, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 11 x^{14} + 7 x^{13} + 29 x^{12} + 139 x^{11} + 527 x^{10} - 2880 x^{9} - 943 x^{8} + 12374 x^{7} - 8494 x^{6} - 7705 x^{5} + 21841 x^{4} - 15244 x^{3} + 19216 x^{2} - 8640 x + 2304 \)

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:  \(11654783511381160590876241=13^{12}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.87$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{6} + \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{7} + \frac{1}{3} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{12} a^{5} - \frac{1}{6} a^{3} - \frac{5}{12} a^{2} + \frac{1}{12} a$, $\frac{1}{56304} a^{14} - \frac{185}{28152} a^{13} + \frac{3637}{56304} a^{12} + \frac{1543}{56304} a^{11} + \frac{1957}{56304} a^{10} + \frac{499}{6256} a^{9} + \frac{6349}{18768} a^{8} - \frac{749}{2346} a^{7} + \frac{10001}{56304} a^{6} - \frac{12925}{28152} a^{5} - \frac{4091}{28152} a^{4} + \frac{3455}{56304} a^{3} - \frac{6319}{56304} a^{2} - \frac{839}{4692} a - \frac{11}{391}$, $\frac{1}{2991652927678511125699454784} a^{15} + \frac{8155953874303483945139}{1495826463839255562849727392} a^{14} + \frac{4443453046180187048666083}{130071866420804831552150208} a^{13} - \frac{69740037440936268036143035}{997217642559503708566484928} a^{12} + \frac{20658435794145070350890669}{332405880853167902855494976} a^{11} - \frac{151049934593576687596332013}{2991652927678511125699454784} a^{10} + \frac{29735063463402667470658381}{997217642559503708566484928} a^{9} - \frac{1784559140484513506490523}{124652205319937963570810616} a^{8} + \frac{125429063353659818976073937}{2991652927678511125699454784} a^{7} + \frac{446659126009109866757976415}{1495826463839255562849727392} a^{6} + \frac{576287822915606746712558129}{1495826463839255562849727392} a^{5} - \frac{176688113444599030616038627}{997217642559503708566484928} a^{4} + \frac{52158984104763695007051297}{332405880853167902855494976} a^{3} - \frac{187438349541053826465245825}{747913231919627781424863696} a^{2} + \frac{26700548291156918299474721}{62326102659968981785405308} a - \frac{1206212104451845523084676}{5193841888330748482117109}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 723660.392097 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

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 $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{377}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{29})\), 4.4.1847677.1 x2, 4.4.63713.1 x2, 4.0.10933.1 x2, 4.0.4901.1 x2, 4.0.2197.1, 4.0.1847677.1, 8.8.3413910296329.1, 8.0.20200652641.1, 8.0.3413910296329.2, 8.0.4059346369.2 x2, 8.0.3413910296329.1 x2

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

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 ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$13.4.3.2$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}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$