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Properties

Label 16.0.33708728269...9277.7
Degree $16$
Signature $[0, 8]$
Discriminant $11^{10}\cdot 37^{9}$
Root discriminant $34.12$
Ramified primes $11, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9959, -25348, 24231, -12097, 10500, -11477, 2523, 2277, 867, -992, -445, 163, 103, 4, -14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 14*x^14 + 4*x^13 + 103*x^12 + 163*x^11 - 445*x^10 - 992*x^9 + 867*x^8 + 2277*x^7 + 2523*x^6 - 11477*x^5 + 10500*x^4 - 12097*x^3 + 24231*x^2 - 25348*x + 9959)
 
gp: K = bnfinit(x^16 - 2*x^15 - 14*x^14 + 4*x^13 + 103*x^12 + 163*x^11 - 445*x^10 - 992*x^9 + 867*x^8 + 2277*x^7 + 2523*x^6 - 11477*x^5 + 10500*x^4 - 12097*x^3 + 24231*x^2 - 25348*x + 9959, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 14 x^{14} + 4 x^{13} + 103 x^{12} + 163 x^{11} - 445 x^{10} - 992 x^{9} + 867 x^{8} + 2277 x^{7} + 2523 x^{6} - 11477 x^{5} + 10500 x^{4} - 12097 x^{3} + 24231 x^{2} - 25348 x + 9959 \)

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:  \(3370872826949590878489277=11^{10}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.12$
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:  $11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{24360665854812052207385015848693} a^{15} - \frac{8070658211193978869468417409967}{24360665854812052207385015848693} a^{14} + \frac{4329939227686239954898288592029}{24360665854812052207385015848693} a^{13} - \frac{2243189085562893579311254813278}{24360665854812052207385015848693} a^{12} + \frac{6119256301617749575882099551041}{24360665854812052207385015848693} a^{11} + \frac{1107652743277225603195675240839}{24360665854812052207385015848693} a^{10} + \frac{5275020087699520349131966298297}{24360665854812052207385015848693} a^{9} - \frac{215448766558864088720838222449}{24360665854812052207385015848693} a^{8} - \frac{4278987735413491313995374305159}{24360665854812052207385015848693} a^{7} + \frac{5657582449308489255992938433344}{24360665854812052207385015848693} a^{6} + \frac{7577450672133859440836855850375}{24360665854812052207385015848693} a^{5} - \frac{2095254749137334117085468996401}{24360665854812052207385015848693} a^{4} + \frac{7301750708214583159266334002996}{24360665854812052207385015848693} a^{3} - \frac{6368392019735408942817925845443}{24360665854812052207385015848693} a^{2} - \frac{8986998318074411973158964751581}{24360665854812052207385015848693} a - \frac{5261363413683627665545588364503}{24360665854812052207385015848693}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 340254.324475 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.4477.1, 8.0.741610573.1

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$11$11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
37Data not computed