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Properties

Label 16.0.68668573773...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 109^{12}$
Root discriminant $112.80$
Ramified primes $5, 109$
Class number $1000$ (GRH)
Class group $[10, 10, 10]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![78074896, -253328120, 460788564, -537558270, 439402015, -239767135, 94786356, -26546400, 5410069, -765600, 113001, -14440, 2665, -330, 54, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 54*x^14 - 330*x^13 + 2665*x^12 - 14440*x^11 + 113001*x^10 - 765600*x^9 + 5410069*x^8 - 26546400*x^7 + 94786356*x^6 - 239767135*x^5 + 439402015*x^4 - 537558270*x^3 + 460788564*x^2 - 253328120*x + 78074896)
 
gp: K = bnfinit(x^16 - 5*x^15 + 54*x^14 - 330*x^13 + 2665*x^12 - 14440*x^11 + 113001*x^10 - 765600*x^9 + 5410069*x^8 - 26546400*x^7 + 94786356*x^6 - 239767135*x^5 + 439402015*x^4 - 537558270*x^3 + 460788564*x^2 - 253328120*x + 78074896, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 54 x^{14} - 330 x^{13} + 2665 x^{12} - 14440 x^{11} + 113001 x^{10} - 765600 x^{9} + 5410069 x^{8} - 26546400 x^{7} + 94786356 x^{6} - 239767135 x^{5} + 439402015 x^{4} - 537558270 x^{3} + 460788564 x^{2} - 253328120 x + 78074896 \)

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:  \(686685737739964474054511962890625=5^{12}\cdot 109^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.80$
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:  $5, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} + \frac{1}{8} a^{9} + \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{376} a^{13} - \frac{5}{376} a^{12} + \frac{7}{376} a^{11} - \frac{1}{376} a^{10} - \frac{61}{376} a^{9} + \frac{83}{376} a^{8} - \frac{81}{376} a^{7} - \frac{17}{376} a^{6} + \frac{87}{376} a^{5} - \frac{1}{376} a^{4} + \frac{23}{188} a^{3} + \frac{7}{94} a^{2} - \frac{23}{94} a$, $\frac{1}{657369120836945517272} a^{14} + \frac{102461189679678653}{82171140104618189659} a^{13} + \frac{13211646389291062833}{657369120836945517272} a^{12} - \frac{524819935275815095}{82171140104618189659} a^{11} + \frac{18478571740586320813}{657369120836945517272} a^{10} + \frac{7414222032597205727}{82171140104618189659} a^{9} + \frac{130286298538534569777}{657369120836945517272} a^{8} - \frac{14869058466404321363}{82171140104618189659} a^{7} - \frac{103218600449907909771}{657369120836945517272} a^{6} + \frac{1259669637747745955}{164342280209236379318} a^{5} - \frac{36284239687036741301}{82171140104618189659} a^{4} - \frac{60124100727263629111}{164342280209236379318} a^{3} - \frac{33330523479642716199}{164342280209236379318} a^{2} - \frac{55092367079815497}{3496644259770986794} a + \frac{15812522612900328}{37198343189053051}$, $\frac{1}{17822942468440926444516582547908097485944} a^{15} + \frac{5516752258483558835}{8911471234220463222258291273954048742972} a^{14} + \frac{1395311721274620026627686284689348243}{8911471234220463222258291273954048742972} a^{13} - \frac{4495164427739247286489170830252696893}{2227867808555115805564572818488512185743} a^{12} - \frac{481278380169951141979788603928664464205}{17822942468440926444516582547908097485944} a^{11} - \frac{138305244021298245188275591541113310738}{2227867808555115805564572818488512185743} a^{10} + \frac{1199838515798165256457124489539500623183}{17822942468440926444516582547908097485944} a^{9} - \frac{717701379859983791811757890193834976675}{4455735617110231611129145636977024371486} a^{8} - \frac{2551580138408327106093911507729559003841}{17822942468440926444516582547908097485944} a^{7} + \frac{553736075958159084427198828065779920380}{2227867808555115805564572818488512185743} a^{6} + \frac{1877058164711910735072037308235666032607}{8911471234220463222258291273954048742972} a^{5} - \frac{1960815097602337860120544207062321048909}{8911471234220463222258291273954048742972} a^{4} + \frac{4534463346275040860416651512756739698941}{17822942468440926444516582547908097485944} a^{3} - \frac{23005188307032559520405320483519175873}{47401442735215229905629208904010897569} a^{2} + \frac{711890111294265156056378908338304979}{2017082669583626804494859953362165854} a - \frac{371860483776611559291690938792308}{21458326272166242601009148440023041}$

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

Class group and class number

$C_{10}\times C_{10}\times C_{10}$, which has order $1000$ (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:  \( -\frac{889511737245090164332804608924191}{17822942468440926444516582547908097485944} a^{15} + \frac{4160847939162500393602732680405887}{17822942468440926444516582547908097485944} a^{14} - \frac{23030646841330133254874102800949335}{8911471234220463222258291273954048742972} a^{13} + \frac{138021013294647423821217749908704279}{8911471234220463222258291273954048742972} a^{12} - \frac{2251508131736902225217525837153986895}{17822942468440926444516582547908097485944} a^{11} + \frac{2984761407764826499798946222888849135}{4455735617110231611129145636977024371486} a^{10} - \frac{95209521587585633565984879408370548663}{17822942468440926444516582547908097485944} a^{9} + \frac{160691981867217052882600524154717928605}{4455735617110231611129145636977024371486} a^{8} - \frac{4543435068965683126099594585535252590875}{17822942468440926444516582547908097485944} a^{7} + \frac{5432737673244987350883977767150476885049}{4455735617110231611129145636977024371486} a^{6} - \frac{18595700866174655666643624944167220151655}{4455735617110231611129145636977024371486} a^{5} + \frac{175840293755469119950590912932828038503657}{17822942468440926444516582547908097485944} a^{4} - \frac{292013130766514339735800064265567583727101}{17822942468440926444516582547908097485944} a^{3} + \frac{3122981019045295605520341456959515098505}{189605770940860919622516835616043590276} a^{2} - \frac{20591620769006675351477834776324824145}{2017082669583626804494859953362165854} a + \frac{78321050699339883197762124863465231}{21458326272166242601009148440023041} \) (order $10$)
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:  \( 29917425.3971 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{109}) \), \(\Q(\sqrt{545}) \), \(\Q(\sqrt{5}, \sqrt{109})\), 4.4.6475145.1 x2, 4.4.32375725.1 x2, 4.0.1485125.1, \(\Q(\zeta_{5})\), 8.8.1048187569275625.1, 8.0.2205596265625.3, 8.0.26204689231890625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$109$109.8.6.1$x^{8} - 5341 x^{4} + 15397776$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
109.8.6.1$x^{8} - 5341 x^{4} + 15397776$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$