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Properties

Label 16.16.1471082571...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $3^{4}\cdot 5^{14}\cdot 29^{14}$
Root discriminant $102.44$
Ramified primes $3, 5, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1223095, 2163460, -3118060, -4046660, 3030830, 2843520, -1450125, -961140, 375226, 164608, -53257, -13424, 3900, 406, -122, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 122*x^14 + 406*x^13 + 3900*x^12 - 13424*x^11 - 53257*x^10 + 164608*x^9 + 375226*x^8 - 961140*x^7 - 1450125*x^6 + 2843520*x^5 + 3030830*x^4 - 4046660*x^3 - 3118060*x^2 + 2163460*x + 1223095)
 
gp: K = bnfinit(x^16 - 2*x^15 - 122*x^14 + 406*x^13 + 3900*x^12 - 13424*x^11 - 53257*x^10 + 164608*x^9 + 375226*x^8 - 961140*x^7 - 1450125*x^6 + 2843520*x^5 + 3030830*x^4 - 4046660*x^3 - 3118060*x^2 + 2163460*x + 1223095, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 122 x^{14} + 406 x^{13} + 3900 x^{12} - 13424 x^{11} - 53257 x^{10} + 164608 x^{9} + 375226 x^{8} - 961140 x^{7} - 1450125 x^{6} + 2843520 x^{5} + 3030830 x^{4} - 4046660 x^{3} - 3118060 x^{2} + 2163460 x + 1223095 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(147108257121214334362554931640625=3^{4}\cdot 5^{14}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $102.44$
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:  $3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a$, $\frac{1}{60965364311030049644290102601805080552} a^{15} - \frac{4214182760059863434609587098332829127}{60965364311030049644290102601805080552} a^{14} + \frac{2757879202368200418272931497680350795}{60965364311030049644290102601805080552} a^{13} - \frac{7612318342938863524913104749903944431}{60965364311030049644290102601805080552} a^{12} - \frac{6965961078017859818499663241898566105}{60965364311030049644290102601805080552} a^{11} + \frac{3417099946702751679787283997482122599}{60965364311030049644290102601805080552} a^{10} - \frac{6110450827167937089415667664469416841}{30482682155515024822145051300902540276} a^{9} + \frac{7236790732786919828094171658817839571}{30482682155515024822145051300902540276} a^{8} + \frac{139490521721998673126288396657887691}{7620670538878756205536262825225635069} a^{7} - \frac{3282651146032372723099797325581593598}{7620670538878756205536262825225635069} a^{6} + \frac{12393718013292905753588541629444037327}{60965364311030049644290102601805080552} a^{5} - \frac{28294944113902967513750075788307827035}{60965364311030049644290102601805080552} a^{4} + \frac{4615655663054492238941429477987431063}{60965364311030049644290102601805080552} a^{3} - \frac{25345916220131078938030884496080203429}{60965364311030049644290102601805080552} a^{2} - \frac{16511356548811583004051163398430407439}{60965364311030049644290102601805080552} a - \frac{5077146523434008373553798941649273143}{60965364311030049644290102601805080552}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $15$
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:  \( 28733573656.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.4.3048625.2, 4.4.3048625.1, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.9294114390625.2

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$29$29.8.7.1$x^{8} - 29$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.1$x^{8} - 29$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$