Properties

Label 16.0.5415791883780096.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{10}\cdot 7^{2}\cdot 13^{4}$
Root discriminant $9.62$
Ramified primes $2, 3, 7, 13$
Class number $1$
Class group Trivial
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![1, -8, 32, -84, 160, -234, 278, -284, 259, -208, 143, -84, 46, -24, 11, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 11*x^14 - 24*x^13 + 46*x^12 - 84*x^11 + 143*x^10 - 208*x^9 + 259*x^8 - 284*x^7 + 278*x^6 - 234*x^5 + 160*x^4 - 84*x^3 + 32*x^2 - 8*x + 1)
gp: K = bnfinit(x^16 - 4*x^15 + 11*x^14 - 24*x^13 + 46*x^12 - 84*x^11 + 143*x^10 - 208*x^9 + 259*x^8 - 284*x^7 + 278*x^6 - 234*x^5 + 160*x^4 - 84*x^3 + 32*x^2 - 8*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 4 x^{15} \) \(\mathstrut +\mathstrut 11 x^{14} \) \(\mathstrut -\mathstrut 24 x^{13} \) \(\mathstrut +\mathstrut 46 x^{12} \) \(\mathstrut -\mathstrut 84 x^{11} \) \(\mathstrut +\mathstrut 143 x^{10} \) \(\mathstrut -\mathstrut 208 x^{9} \) \(\mathstrut +\mathstrut 259 x^{8} \) \(\mathstrut -\mathstrut 284 x^{7} \) \(\mathstrut +\mathstrut 278 x^{6} \) \(\mathstrut -\mathstrut 234 x^{5} \) \(\mathstrut +\mathstrut 160 x^{4} \) \(\mathstrut -\mathstrut 84 x^{3} \) \(\mathstrut +\mathstrut 32 x^{2} \) \(\mathstrut -\mathstrut 8 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(5415791883780096=2^{16}\cdot 3^{10}\cdot 7^{2}\cdot 13^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.62$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{26911} a^{15} + \frac{13101}{26911} a^{14} - \frac{3564}{26911} a^{13} + \frac{11252}{26911} a^{12} + \frac{12137}{26911} a^{11} + \frac{11291}{26911} a^{10} + \frac{12020}{26911} a^{9} + \frac{11809}{26911} a^{8} - \frac{7957}{26911} a^{7} + \frac{3356}{26911} a^{6} + \frac{8084}{26911} a^{5} - \frac{8021}{26911} a^{4} - \frac{679}{26911} a^{3} + \frac{9162}{26911} a^{2} - \frac{520}{1583} a + \frac{3647}{26911}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( -\frac{208514}{26911} a^{15} + \frac{801026}{26911} a^{14} - \frac{2102427}{26911} a^{13} + \frac{4476322}{26911} a^{12} - \frac{8393299}{26911} a^{11} + \frac{15208887}{26911} a^{10} - \frac{25628478}{26911} a^{9} + \frac{36102325}{26911} a^{8} - \frac{43055585}{26911} a^{7} + \frac{45634805}{26911} a^{6} - \frac{43168113}{26911} a^{5} + \frac{34337491}{26911} a^{4} - \frac{21149811}{26911} a^{3} + \frac{9452383}{26911} a^{2} - \frac{166520}{1583} a + \frac{457967}{26911} \) (order $12$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{190008}{26911} a^{15} - \frac{726200}{26911} a^{14} + \frac{1937484}{26911} a^{13} - \frac{4145584}{26911} a^{12} + \frac{7820052}{26911} a^{11} - \frac{14200511}{26911} a^{10} + \frac{23964202}{26911} a^{9} - \frac{34023301}{26911} a^{8} + \frac{41140154}{26911} a^{7} - \frac{43961871}{26911} a^{6} + \frac{41872130}{26911} a^{5} - \frac{33803721}{26911} a^{4} + \frac{21470880}{26911} a^{3} - \frac{9949453}{26911} a^{2} + \frac{180830}{1583} a - \frac{483472}{26911} \),  \( \frac{330451}{26911} a^{15} - \frac{1118103}{26911} a^{14} + \frac{2938939}{26911} a^{13} - \frac{6086282}{26911} a^{12} + \frac{11359344}{26911} a^{11} - \frac{20567180}{26911} a^{10} + \frac{34215123}{26911} a^{9} - \frac{47001035}{26911} a^{8} + \frac{55511863}{26911} a^{7} - \frac{58028870}{26911} a^{6} + \frac{54217312}{26911} a^{5} - \frac{41983508}{26911} a^{4} + \frac{25142563}{26911} a^{3} - \frac{10821304}{26911} a^{2} + \frac{182175}{1583} a - \frac{458003}{26911} \),  \( \frac{7255}{26911} a^{15} + \frac{51925}{26911} a^{14} - \frac{129904}{26911} a^{13} + \frac{362040}{26911} a^{12} - \frac{698543}{26911} a^{11} + \frac{1263938}{26911} a^{10} - \frac{2354708}{26911} a^{9} + \frac{3891766}{26911} a^{8} - \frac{4821009}{26911} a^{7} + \frac{5429347}{26911} a^{6} - \frac{5371849}{26911} a^{5} + \frac{4725563}{26911} a^{4} - \frac{3096197}{26911} a^{3} + \frac{1426423}{26911} a^{2} - \frac{24056}{1583} a + \frac{32383}{26911} \),  \( \frac{228015}{26911} a^{15} - \frac{811459}{26911} a^{14} + \frac{2138887}{26911} a^{13} - \frac{4510275}{26911} a^{12} + \frac{8448513}{26911} a^{11} - \frac{15343453}{26911} a^{10} + \frac{25689510}{26911} a^{9} - \frac{35823333}{26911} a^{8} + \frac{42866577}{26911} a^{7} - \frac{45475535}{26911} a^{6} + \frac{42927360}{26911} a^{5} - \frac{34079170}{26911} a^{4} + \frac{21202666}{26911} a^{3} - \frac{9715460}{26911} a^{2} + \frac{176196}{1583} a - \frac{490504}{26911} \),  \( \frac{195184}{26911} a^{15} - \frac{596589}{26911} a^{14} + \frac{1547501}{26911} a^{13} - \frac{3144929}{26911} a^{12} + \frac{5812565}{26911} a^{11} - \frac{10549091}{26911} a^{10} + \frac{17314473}{26911} a^{9} - \frac{23088932}{26911} a^{8} + \frac{26973366}{26911} a^{7} - \frac{27990587}{26911} a^{6} + \frac{25721709}{26911} a^{5} - \frac{19372448}{26911} a^{4} + \frac{11309359}{26911} a^{3} - \frac{4831033}{26911} a^{2} + \frac{80681}{1583} a - \frac{202101}{26911} \),  \( \frac{76625}{26911} a^{15} - \frac{158374}{26911} a^{14} + \frac{378082}{26911} a^{13} - \frac{635982}{26911} a^{12} + \frac{1083727}{26911} a^{11} - \frac{1926456}{26911} a^{10} + \frac{2802269}{26911} a^{9} - \frac{2520473}{26911} a^{8} + \frac{1928083}{26911} a^{7} - \frac{1003723}{26911} a^{6} - \frac{27809}{26911} a^{5} + \frac{1598953}{26911} a^{4} - \frac{2269936}{26911} a^{3} + \frac{1787119}{26911} a^{2} - \frac{46797}{1583} a + \frac{195928}{26911} \),  \( \frac{169657}{26911} a^{15} - \frac{602819}{26911} a^{14} + \frac{1593460}{26911} a^{13} - \frac{3342007}{26911} a^{12} + \frac{6275196}{26911} a^{11} - \frac{11364968}{26911} a^{10} + \frac{19041459}{26911} a^{9} - \frac{26524861}{26911} a^{8} + \frac{31649991}{26911} a^{7} - \frac{33464419}{26911} a^{6} + \frac{31608498}{26911} a^{5} - \frac{24929846}{26911} a^{4} + \frac{15455802}{26911} a^{3} - \frac{6951875}{26911} a^{2} + \frac{125590}{1583} a - \frac{348476}{26911} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 63.1744988972 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), 4.0.1872.1, 4.0.117.1, \(\Q(\zeta_{12})\), 8.0.3504384.2

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$