Properties

Label 16.0.4897760256000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{14}\cdot 5^{6}$
Root discriminant $9.56$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois Group $(C_4\times C_8):C_2$ (as 16T114)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 35, -102, 218, -352, 433, -396, 253, -90, -8, 28, -10, -6, 8, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 6*x^13 - 10*x^12 + 28*x^11 - 8*x^10 - 90*x^9 + 253*x^8 - 396*x^7 + 433*x^6 - 352*x^5 + 218*x^4 - 102*x^3 + 35*x^2 - 8*x + 1)
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 6*x^13 - 10*x^12 + 28*x^11 - 8*x^10 - 90*x^9 + 253*x^8 - 396*x^7 + 433*x^6 - 352*x^5 + 218*x^4 - 102*x^3 + 35*x^2 - 8*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 4 x^{15} \) \(\mathstrut +\mathstrut 8 x^{14} \) \(\mathstrut -\mathstrut 6 x^{13} \) \(\mathstrut -\mathstrut 10 x^{12} \) \(\mathstrut +\mathstrut 28 x^{11} \) \(\mathstrut -\mathstrut 8 x^{10} \) \(\mathstrut -\mathstrut 90 x^{9} \) \(\mathstrut +\mathstrut 253 x^{8} \) \(\mathstrut -\mathstrut 396 x^{7} \) \(\mathstrut +\mathstrut 433 x^{6} \) \(\mathstrut -\mathstrut 352 x^{5} \) \(\mathstrut +\mathstrut 218 x^{4} \) \(\mathstrut -\mathstrut 102 x^{3} \) \(\mathstrut +\mathstrut 35 x^{2} \) \(\mathstrut -\mathstrut 8 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(4897760256000000=2^{16}\cdot 3^{14}\cdot 5^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.56$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{31} a^{15} - \frac{1}{31} a^{14} + \frac{5}{31} a^{13} + \frac{9}{31} a^{12} - \frac{14}{31} a^{11} - \frac{14}{31} a^{10} + \frac{12}{31} a^{9} + \frac{8}{31} a^{8} - \frac{2}{31} a^{7} + \frac{1}{31} a^{6} + \frac{2}{31} a^{5} - \frac{5}{31} a^{4} - \frac{14}{31} a^{3} + \frac{11}{31} a^{2} + \frac{6}{31} a + \frac{10}{31}$

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:  \( -\frac{265}{31} a^{15} + \frac{575}{31} a^{14} - \frac{643}{31} a^{13} - \frac{711}{31} a^{12} + \frac{2904}{31} a^{11} - \frac{1684}{31} a^{10} - \frac{5846}{31} a^{9} + \frac{18154}{31} a^{8} - \frac{26719}{31} a^{7} + \frac{23481}{31} a^{6} - \frac{13643}{31} a^{5} + \frac{3712}{31} a^{4} + \frac{517}{31} a^{3} - \frac{1427}{31} a^{2} + \frac{611}{31} a - \frac{201}{31} \) (order $12$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{405}{31} a^{15} - \frac{1738}{31} a^{14} + \frac{3420}{31} a^{13} - \frac{2462}{31} a^{12} - \frac{4802}{31} a^{11} + \frac{12775}{31} a^{10} - \frac{3076}{31} a^{9} - \frac{40625}{31} a^{8} + \frac{110046}{31} a^{7} - \frac{165538}{31} a^{6} + \frac{170938}{31} a^{5} - \frac{128970}{31} a^{4} + \frac{71768}{31} a^{3} - \frac{29025}{31} a^{2} + \frac{7762}{31} a - \frac{1158}{31} \),  \( \frac{1779}{31} a^{15} - \frac{5375}{31} a^{14} + \frac{8089}{31} a^{13} - \frac{140}{31} a^{12} - \frac{21806}{31} a^{11} + \frac{28352}{31} a^{10} + \frac{24293}{31} a^{9} - \frac{149820}{31} a^{8} + \frac{290787}{31} a^{7} - \frac{346382}{31} a^{6} + \frac{290928}{31} a^{5} - \frac{176636}{31} a^{4} + \frac{78417}{31} a^{3} - \frac{23273}{31} a^{2} + \frac{4102}{31} a - \frac{128}{31} \),  \( \frac{333}{31} a^{15} - \frac{2100}{31} a^{14} + \frac{4734}{31} a^{13} - \frac{4784}{31} a^{12} - \frac{4290}{31} a^{11} + \frac{18619}{31} a^{10} - \frac{11938}{31} a^{9} - \frac{43898}{31} a^{8} + \frac{145127}{31} a^{7} - \frac{237344}{31} a^{6} + \frac{256292}{31} a^{5} - \frac{199631}{31} a^{4} + \frac{113479}{31} a^{3} - \frac{47053}{31} a^{2} + \frac{12879}{31} a - \frac{2033}{31} \),  \( \frac{375}{31} a^{15} - \frac{2359}{31} a^{14} + \frac{5161}{31} a^{13} - \frac{4964}{31} a^{12} - \frac{5312}{31} a^{11} + \frac{20604}{31} a^{10} - \frac{11496}{31} a^{9} - \frac{50878}{31} a^{8} + \frac{159706}{31} a^{7} - \frac{254569}{31} a^{6} + \frac{268559}{31} a^{5} - \frac{205266}{31} a^{4} + \frac{114627}{31} a^{3} - \frac{46808}{31} a^{2} + \frac{12511}{31} a - \frac{1954}{31} \),  \( \frac{640}{31} a^{15} - \frac{1973}{31} a^{14} + \frac{3045}{31} a^{13} - \frac{285}{31} a^{12} - \frac{7751}{31} a^{11} + \frac{10663}{31} a^{10} + \frac{7897}{31} a^{9} - \frac{54090}{31} a^{8} + \frac{108088}{31} a^{7} - \frac{132567}{31} a^{6} + \frac{115484}{31} a^{5} - \frac{73911}{31} a^{4} + \frac{35401}{31} a^{3} - \frac{11994}{31} a^{2} + \frac{2693}{31} a - \frac{265}{31} \),  \( \frac{12}{31} a^{15} + \frac{577}{31} a^{14} - \frac{1738}{31} a^{13} + \frac{2619}{31} a^{12} + \frac{49}{31} a^{11} - \frac{7298}{31} a^{10} + \frac{9227}{31} a^{9} + \frac{8187}{31} a^{8} - \frac{48787}{31} a^{7} + \frac{93105}{31} a^{6} - \frac{108445}{31} a^{5} + \frac{88724}{31} a^{4} - \frac{51876}{31} a^{3} + \frac{22018}{31} a^{2} - \frac{6066}{31} a + \frac{957}{31} \),  \( \frac{116}{31} a^{15} + \frac{535}{31} a^{14} - \frac{2117}{31} a^{13} + \frac{3958}{31} a^{12} - \frac{1438}{31} a^{11} - \frac{8909}{31} a^{10} + \frac{15404}{31} a^{9} + \frac{2385}{31} a^{8} - \frac{54885}{31} a^{7} + \frac{120365}{31} a^{6} - \frac{151048}{31} a^{5} + \frac{131232}{31} a^{4} - \frac{81232}{31} a^{3} + \frac{36399}{31} a^{2} - \frac{10588}{31} a + \frac{1749}{31} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 59.5214550029 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$(C_4\times C_8):C_2$ (as 16T114):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 64
The 28 conjugacy class representatives for $(C_4\times C_8):C_2$
Character table for $(C_4\times C_8):C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 8.0.4665600.1

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

Sibling fields

Degree 16 sibling: 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 R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
$5$5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
5.8.6.4$x^{8} - 5 x^{4} + 50$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$