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Properties

Label 16.4.41239710384...1376.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{40}\cdot 41^{6}\cdot 281^{2}$
Root discriminant $46.07$
Ramified primes $2, 41, 281$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1127

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![332612, 1676384, 556960, -381712, -388460, 16560, 107288, 31472, -12086, -8720, -552, 1048, 288, -52, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 - 52*x^13 + 288*x^12 + 1048*x^11 - 552*x^10 - 8720*x^9 - 12086*x^8 + 31472*x^7 + 107288*x^6 + 16560*x^5 - 388460*x^4 - 381712*x^3 + 556960*x^2 + 1676384*x + 332612)
 
gp: K = bnfinit(x^16 - 32*x^14 - 52*x^13 + 288*x^12 + 1048*x^11 - 552*x^10 - 8720*x^9 - 12086*x^8 + 31472*x^7 + 107288*x^6 + 16560*x^5 - 388460*x^4 - 381712*x^3 + 556960*x^2 + 1676384*x + 332612, 1)
 

Normalized defining polynomial

\( x^{16} - 32 x^{14} - 52 x^{13} + 288 x^{12} + 1048 x^{11} - 552 x^{10} - 8720 x^{9} - 12086 x^{8} + 31472 x^{7} + 107288 x^{6} + 16560 x^{5} - 388460 x^{4} - 381712 x^{3} + 556960 x^{2} + 1676384 x + 332612 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(412397103845080712794341376=2^{40}\cdot 41^{6}\cdot 281^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.07$
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:  $2, 41, 281$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{13611231618306615848271181670580754014860998} a^{15} - \frac{592981206710391659332961320327945394662874}{6805615809153307924135590835290377007430499} a^{14} - \frac{1583266413972375539378978193998065207767243}{6805615809153307924135590835290377007430499} a^{13} - \frac{680742738601647839647657317135657474401883}{13611231618306615848271181670580754014860998} a^{12} + \frac{1160796312253079863528072443193203663601583}{13611231618306615848271181670580754014860998} a^{11} + \frac{1049615291899279351817600584401197519428121}{6805615809153307924135590835290377007430499} a^{10} - \frac{907084378670035901357320439320804691536308}{6805615809153307924135590835290377007430499} a^{9} + \frac{3009084829318225026709987072459214831106921}{13611231618306615848271181670580754014860998} a^{8} + \frac{1496787719699078456936744741840095902667041}{6805615809153307924135590835290377007430499} a^{7} - \frac{563154603701341783244839298870990170423611}{6805615809153307924135590835290377007430499} a^{6} + \frac{2286356764881313596974935514094007583404851}{6805615809153307924135590835290377007430499} a^{5} - \frac{1178062691619703661396003517252589769099877}{6805615809153307924135590835290377007430499} a^{4} + \frac{1409895715263962427872266062676178314552150}{6805615809153307924135590835290377007430499} a^{3} + \frac{561964224035539518336811866309894832611539}{6805615809153307924135590835290377007430499} a^{2} + \frac{3154422014653127134520803557028208642861479}{6805615809153307924135590835290377007430499} a + \frac{436269641362797770877673934477533976414155}{6805615809153307924135590835290377007430499}$

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:  $9$
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:  \( 7595822.35126 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1127:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 58 conjugacy class representatives for t16n1127 are not computed
Character table for t16n1127 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.4.110166016.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
281Data not computed