Properties

Label 16.0.6158959248447369.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 7^{4}\cdot 13^{6}$
Root discriminant $9.70$
Ramified primes $3, 7, 13$
Class number $1$
Class group Trivial
Galois Group $C_4.C_2^2:D_4$ (as 16T211)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 17, -27, 23, 3, -34, 45, -29, 3, 13, -18, 17, -12, 7, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 17*x^12 - 18*x^11 + 13*x^10 + 3*x^9 - 29*x^8 + 45*x^7 - 34*x^6 + 3*x^5 + 23*x^4 - 27*x^3 + 17*x^2 - 6*x + 1)
gp: K = bnfinit(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 17*x^12 - 18*x^11 + 13*x^10 + 3*x^9 - 29*x^8 + 45*x^7 - 34*x^6 + 3*x^5 + 23*x^4 - 27*x^3 + 17*x^2 - 6*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 3 x^{15} \) \(\mathstrut +\mathstrut 7 x^{14} \) \(\mathstrut -\mathstrut 12 x^{13} \) \(\mathstrut +\mathstrut 17 x^{12} \) \(\mathstrut -\mathstrut 18 x^{11} \) \(\mathstrut +\mathstrut 13 x^{10} \) \(\mathstrut +\mathstrut 3 x^{9} \) \(\mathstrut -\mathstrut 29 x^{8} \) \(\mathstrut +\mathstrut 45 x^{7} \) \(\mathstrut -\mathstrut 34 x^{6} \) \(\mathstrut +\mathstrut 3 x^{5} \) \(\mathstrut +\mathstrut 23 x^{4} \) \(\mathstrut -\mathstrut 27 x^{3} \) \(\mathstrut +\mathstrut 17 x^{2} \) \(\mathstrut -\mathstrut 6 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(6158959248447369=3^{12}\cdot 7^{4}\cdot 13^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.70$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{258287} a^{15} + \frac{30004}{258287} a^{14} - \frac{58447}{258287} a^{13} - \frac{50411}{258287} a^{12} + \frac{104099}{258287} a^{11} - \frac{24303}{258287} a^{10} - \frac{115907}{258287} a^{9} + \frac{71396}{258287} a^{8} - \frac{110922}{258287} a^{7} + \frac{108160}{258287} a^{6} - \frac{77356}{258287} a^{5} + \frac{3780}{258287} a^{4} + \frac{38490}{258287} a^{3} - \frac{90061}{258287} a^{2} - \frac{3529}{258287} a + \frac{2961}{258287}$

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:  \( -\frac{2168}{1427} a^{15} + \frac{6831}{1427} a^{14} - \frac{15920}{1427} a^{13} + \frac{27085}{1427} a^{12} - \frac{37976}{1427} a^{11} + \frac{39739}{1427} a^{10} - \frac{26875}{1427} a^{9} - \frac{9827}{1427} a^{8} + \frac{69352}{1427} a^{7} - \frac{106130}{1427} a^{6} + \frac{76691}{1427} a^{5} + \frac{4502}{1427} a^{4} - \frac{63856}{1427} a^{3} + \frac{62907}{1427} a^{2} - \frac{32096}{1427} a + \frac{7760}{1427} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{742714}{258287} a^{15} - \frac{2161226}{258287} a^{14} + \frac{4765237}{258287} a^{13} - \frac{7937118}{258287} a^{12} + \frac{10743873}{258287} a^{11} - \frac{10639401}{258287} a^{10} + \frac{6471242}{258287} a^{9} + \frac{4620236}{258287} a^{8} - \frac{21796896}{258287} a^{7} + \frac{29226505}{258287} a^{6} - \frac{16812559}{258287} a^{5} - \frac{5028223}{258287} a^{4} + \frac{17936790}{258287} a^{3} - \frac{15445236}{258287} a^{2} + \frac{7032519}{258287} a - \frac{1687373}{258287} \),  \( \frac{511103}{258287} a^{15} - \frac{914500}{258287} a^{14} + \frac{2070527}{258287} a^{13} - \frac{2893092}{258287} a^{12} + \frac{3613224}{258287} a^{11} - \frac{2638962}{258287} a^{10} + \frac{807473}{258287} a^{9} + \frac{4313307}{258287} a^{8} - \frac{9935094}{258287} a^{7} + \frac{7482480}{258287} a^{6} - \frac{634291}{258287} a^{5} - \frac{4408299}{258287} a^{4} + \frac{5349142}{258287} a^{3} - \frac{2929022}{258287} a^{2} + \frac{1485356}{258287} a - \frac{185837}{258287} \),  \( \frac{51764}{258287} a^{15} + \frac{47325}{258287} a^{14} - \frac{134877}{258287} a^{13} + \frac{515131}{258287} a^{12} - \frac{835906}{258287} a^{11} + \frac{1386920}{258287} a^{10} - \frac{1352660}{258287} a^{9} + \frac{1205296}{258287} a^{8} + \frac{211889}{258287} a^{7} - \frac{2934216}{258287} a^{6} + \frac{4358256}{258287} a^{5} - \frac{2438209}{258287} a^{4} - \frac{1320993}{258287} a^{3} + \frac{3003903}{258287} a^{2} - \frac{1874256}{258287} a + \frac{625587}{258287} \),  \( \frac{198149}{258287} a^{15} - \frac{762431}{258287} a^{14} + \frac{1665912}{258287} a^{13} - \frac{2997245}{258287} a^{12} + \frac{4187236}{258287} a^{11} - \frac{4503198}{258287} a^{10} + \frac{3123341}{258287} a^{9} + \frac{667014}{258287} a^{8} - \frac{7383149}{258287} a^{7} + \frac{12054930}{258287} a^{6} - \frac{8495500}{258287} a^{5} - \frac{29080}{258287} a^{4} + \frac{6771936}{258287} a^{3} - \frac{7163721}{258287} a^{2} + \frac{3531106}{258287} a - \frac{883736}{258287} \),  \( \frac{151638}{258287} a^{15} - \frac{237240}{258287} a^{14} + \frac{590506}{258287} a^{13} - \frac{736027}{258287} a^{12} + \frac{929018}{258287} a^{11} - \frac{535972}{258287} a^{10} + \frac{8110}{258287} a^{9} + \frac{1538478}{258287} a^{8} - \frac{2923666}{258287} a^{7} + \frac{2007876}{258287} a^{6} - \frac{5023}{258287} a^{5} - \frac{1755222}{258287} a^{4} + \frac{2101577}{258287} a^{3} - \frac{777941}{258287} a^{2} + \frac{40162}{258287} a + \frac{355599}{258287} \),  \( \frac{624254}{258287} a^{15} - \frac{1631085}{258287} a^{14} + \frac{3722387}{258287} a^{13} - \frac{6037489}{258287} a^{12} + \frac{8247991}{258287} a^{11} - \frac{7990053}{258287} a^{10} + \frac{4986107}{258287} a^{9} + \frac{3883030}{258287} a^{8} - \frac{16645587}{258287} a^{7} + \frac{21687504}{258287} a^{6} - \frac{12852680}{258287} a^{5} - \frac{2871069}{258287} a^{4} + \frac{13044348}{258287} a^{3} - \frac{12005980}{258287} a^{2} + \frac{6136345}{258287} a - \frac{1435400}{258287} \),  \( \frac{744774}{258287} a^{15} - \frac{1825292}{258287} a^{14} + \frac{3951298}{258287} a^{13} - \frac{6144395}{258287} a^{12} + \frac{7968446}{258287} a^{11} - \frac{6979885}{258287} a^{10} + \frac{3260566}{258287} a^{9} + \frac{6280415}{258287} a^{8} - \frac{19129351}{258287} a^{7} + \frac{20869240}{258287} a^{6} - \frac{7504508}{258287} a^{5} - \frac{8347764}{258287} a^{4} + \frac{14316063}{258287} a^{3} - \frac{9063655}{258287} a^{2} + \frac{3120510}{258287} a - \frac{495166}{258287} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 33.0234170129 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_4.C_2^2:D_4$ (as 16T211):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.C_2^2:D_4$
Character table for $C_4.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.117.1, 4.0.189.1, 4.0.2457.1, 8.0.1601613.1, 8.0.8719893.1, 8.0.6036849.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
3Data not computed
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$13$13.4.0.1$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}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$