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Properties

Label 16.0.72740373276...3616.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 17^{4}\cdot 89^{2}$
Root discriminant $20.13$
Ramified primes $2, 17, 89$
Class number $2$
Class group $[2]$
Galois group $C_2^3.C_2^4.C_2$ (as 16T595)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12434, 20808, 19528, 18096, 9858, 4132, 1712, -464, 155, -340, 48, -88, 6, -16, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 - 16*x^13 + 6*x^12 - 88*x^11 + 48*x^10 - 340*x^9 + 155*x^8 - 464*x^7 + 1712*x^6 + 4132*x^5 + 9858*x^4 + 18096*x^3 + 19528*x^2 + 20808*x + 12434)
 
gp: K = bnfinit(x^16 + 4*x^14 - 16*x^13 + 6*x^12 - 88*x^11 + 48*x^10 - 340*x^9 + 155*x^8 - 464*x^7 + 1712*x^6 + 4132*x^5 + 9858*x^4 + 18096*x^3 + 19528*x^2 + 20808*x + 12434, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{14} - 16 x^{13} + 6 x^{12} - 88 x^{11} + 48 x^{10} - 340 x^{9} + 155 x^{8} - 464 x^{7} + 1712 x^{6} + 4132 x^{5} + 9858 x^{4} + 18096 x^{3} + 19528 x^{2} + 20808 x + 12434 \)

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:  \(727403732765419503616=2^{40}\cdot 17^{4}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.13$
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:  $2, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1513} a^{14} - \frac{737}{1513} a^{13} - \frac{751}{1513} a^{12} - \frac{675}{1513} a^{11} - \frac{677}{1513} a^{10} + \frac{665}{1513} a^{9} - \frac{263}{1513} a^{8} + \frac{251}{1513} a^{7} + \frac{642}{1513} a^{6} + \frac{582}{1513} a^{5} + \frac{636}{1513} a^{4} - \frac{400}{1513} a^{3} + \frac{227}{1513} a^{2} - \frac{729}{1513} a + \frac{660}{1513}$, $\frac{1}{311655991770405772116917} a^{15} + \frac{69666871629391062641}{311655991770405772116917} a^{14} + \frac{112744250556910212528040}{311655991770405772116917} a^{13} - \frac{5831516652352333104279}{16402946935284514321943} a^{12} + \frac{49695607442803167734349}{311655991770405772116917} a^{11} - \frac{7151073643981001926377}{16402946935284514321943} a^{10} - \frac{40836902347960650396336}{311655991770405772116917} a^{9} - \frac{6097295401826758812186}{311655991770405772116917} a^{8} + \frac{25742115151646314064254}{311655991770405772116917} a^{7} + \frac{31554986938089790456919}{311655991770405772116917} a^{6} - \frac{64403833536508792852743}{311655991770405772116917} a^{5} + \frac{56777139926694143954476}{311655991770405772116917} a^{4} - \frac{7353048267858674693024}{311655991770405772116917} a^{3} + \frac{149881680037315604856876}{311655991770405772116917} a^{2} - \frac{115038476563241897235060}{311655991770405772116917} a - \frac{113213382722461855302557}{311655991770405772116917}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{2213366133064977712}{1078394435191715474453} a^{15} - \frac{754448864967088132}{1078394435191715474453} a^{14} + \frac{1721210159850544437}{1078394435191715474453} a^{13} + \frac{1102345906908418222}{56757601852195551287} a^{12} + \frac{16289264705939833443}{1078394435191715474453} a^{11} + \frac{2583577533688114232}{56757601852195551287} a^{10} + \frac{99477813865805997648}{1078394435191715474453} a^{9} + \frac{154870265587598325641}{1078394435191715474453} a^{8} + \frac{445727107190810635117}{1078394435191715474453} a^{7} - \frac{1111610723384902140380}{1078394435191715474453} a^{6} - \frac{2002179113749117669977}{1078394435191715474453} a^{5} - \frac{9129765378199379332298}{1078394435191715474453} a^{4} - \frac{14883855877624665234462}{1078394435191715474453} a^{3} - \frac{17225691566802637411719}{1078394435191715474453} a^{2} - \frac{21028799635502701539956}{1078394435191715474453} a - \frac{10462418536230486030719}{1078394435191715474453} \) (order $8$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 29074.9950544 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T595):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), 4.4.4352.1, \(\Q(\zeta_{8})\), 4.0.1088.2, 8.0.18939904.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.20.53$x^{8} + 4 x^{5} + 2 x^{4} + 2$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2}$
2.8.20.53$x^{8} + 4 x^{5} + 2 x^{4} + 2$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$