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Properties

Label 16.0.14106298732...0561.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 61^{12}$
Root discriminant $49.76$
Ramified primes $3, 61$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![515524, 621070, 328913, 8304, -12802, 65558, 84259, 35484, 4582, -1635, 1411, 484, -79, 12, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - x^14 + 12*x^13 - 79*x^12 + 484*x^11 + 1411*x^10 - 1635*x^9 + 4582*x^8 + 35484*x^7 + 84259*x^6 + 65558*x^5 - 12802*x^4 + 8304*x^3 + 328913*x^2 + 621070*x + 515524)
 
gp: K = bnfinit(x^16 - x^15 - x^14 + 12*x^13 - 79*x^12 + 484*x^11 + 1411*x^10 - 1635*x^9 + 4582*x^8 + 35484*x^7 + 84259*x^6 + 65558*x^5 - 12802*x^4 + 8304*x^3 + 328913*x^2 + 621070*x + 515524, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - x^{14} + 12 x^{13} - 79 x^{12} + 484 x^{11} + 1411 x^{10} - 1635 x^{9} + 4582 x^{8} + 35484 x^{7} + 84259 x^{6} + 65558 x^{5} - 12802 x^{4} + 8304 x^{3} + 328913 x^{2} + 621070 x + 515524 \)

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:  \(1410629873249683485564270561=3^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.76$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{15} a^{8} - \frac{2}{15} a^{7} - \frac{1}{15} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{2}{15} a^{2} - \frac{4}{15} a - \frac{2}{15}$, $\frac{1}{15} a^{9} - \frac{1}{3} a^{7} - \frac{7}{15} a^{6} - \frac{1}{3} a^{4} - \frac{1}{5} a^{3} + \frac{1}{3} a - \frac{4}{15}$, $\frac{1}{15} a^{10} - \frac{2}{15} a^{7} - \frac{1}{3} a^{6} + \frac{2}{15} a^{4} - \frac{1}{3} a^{3} + \frac{2}{5} a + \frac{1}{3}$, $\frac{1}{15} a^{11} + \frac{2}{5} a^{7} - \frac{2}{15} a^{6} + \frac{7}{15} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{5} a - \frac{4}{15}$, $\frac{1}{1830} a^{12} + \frac{13}{1830} a^{11} - \frac{9}{610} a^{10} - \frac{1}{366} a^{9} + \frac{14}{915} a^{8} - \frac{283}{610} a^{7} + \frac{54}{305} a^{6} - \frac{49}{1830} a^{5} + \frac{167}{610} a^{4} + \frac{10}{183} a^{3} + \frac{51}{305} a^{2} + \frac{139}{610} a - \frac{173}{915}$, $\frac{1}{1830} a^{13} + \frac{8}{305} a^{11} - \frac{2}{183} a^{10} - \frac{29}{1830} a^{9} + \frac{7}{1830} a^{8} + \frac{249}{610} a^{7} + \frac{25}{122} a^{6} + \frac{203}{915} a^{5} + \frac{35}{366} a^{4} + \frac{296}{915} a^{3} - \frac{511}{1830} a^{2} + \frac{91}{366} a - \frac{26}{183}$, $\frac{1}{40324050} a^{14} + \frac{4382}{20162025} a^{13} - \frac{59}{1550925} a^{12} + \frac{13948}{806481} a^{11} - \frac{651217}{40324050} a^{10} + \frac{194059}{40324050} a^{9} + \frac{1162037}{40324050} a^{8} + \frac{8615003}{40324050} a^{7} - \frac{7595752}{20162025} a^{6} + \frac{3737867}{8064810} a^{5} - \frac{4519027}{20162025} a^{4} + \frac{10377203}{40324050} a^{3} + \frac{47459}{620370} a^{2} - \frac{1962761}{4032405} a + \frac{3980389}{20162025}$, $\frac{1}{2690665782431778759759541650} a^{15} - \frac{24044583450911049139}{2690665782431778759759541650} a^{14} - \frac{76608541415091963611857}{896888594143926253253180550} a^{13} + \frac{41496638555555647467187}{2690665782431778759759541650} a^{12} + \frac{10974663618901294831913054}{1345332891215889379879770825} a^{11} - \frac{1154627366665467374925567}{59792572942928416883545370} a^{10} + \frac{824226958096666442373353}{89688859414392625325318055} a^{9} - \frac{693053320209080697784523}{44109275121832438684582650} a^{8} + \frac{8637690668871617924735939}{896888594143926253253180550} a^{7} - \frac{74893196890301046586676219}{1345332891215889379879770825} a^{6} + \frac{548881512274645229839372183}{1345332891215889379879770825} a^{5} + \frac{10412802637967095373782681}{179377718828785250650636110} a^{4} - \frac{17394044613686670833395373}{149481432357321042208863425} a^{3} - \frac{125436885388364756908265338}{269066578243177875975954165} a^{2} - \frac{160355932348610101823999}{2450515284546246593587925} a + \frac{567629619904720657841267}{3747445379431446740612175}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{29105677301655691}{130115855816614863376350} a^{15} + \frac{257933919271321933}{65057927908307431688175} a^{14} - \frac{284088407104872946}{21685975969435810562725} a^{13} + \frac{2849462318596217147}{130115855816614863376350} a^{12} + \frac{204888063937102673}{21685975969435810562725} a^{11} - \frac{3985563973265440427}{13011585581661486337635} a^{10} + \frac{13894166299911984212}{4337195193887162112545} a^{9} - \frac{7231006898509041553}{130115855816614863376350} a^{8} - \frac{406192812673361319851}{43371951938871621125450} a^{7} + \frac{6855112743307157780297}{130115855816614863376350} a^{6} + \frac{11118540052797955381901}{130115855816614863376350} a^{5} + \frac{2390379380427262643473}{13011585581661486337635} a^{4} - \frac{2762666157668337066803}{43371951938871621125450} a^{3} + \frac{70999688606752367062}{867439038777432422509} a^{2} + \frac{78160357599159019556953}{130115855816614863376350} a + \frac{91892297875611360759}{60406618299264096275} \) (order $6$)
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:  \( 8147753.36594 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

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 $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{-183}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{-3}) \), 4.0.2042829.1 x2, 4.2.680943.1 x2, \(\Q(\sqrt{-3}, \sqrt{61})\), 8.0.615710703429.1 x2, 8.0.37558352909169.1 x2, 8.0.4173150323241.2

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$61$61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$