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Properties

Label 16.0.32790240335...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 7^{8}\cdot 13^{12}$
Root discriminant $60.57$
Ramified primes $5, 7, 13$
Class number $160$ (GRH)
Class group $[4, 40]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![707281, -536558, 992380, -1221277, 1741879, 1023646, 672889, 318736, 219711, 38603, 14056, 2717, 664, -71, 25, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 25*x^14 - 71*x^13 + 664*x^12 + 2717*x^11 + 14056*x^10 + 38603*x^9 + 219711*x^8 + 318736*x^7 + 672889*x^6 + 1023646*x^5 + 1741879*x^4 - 1221277*x^3 + 992380*x^2 - 536558*x + 707281)
 
gp: K = bnfinit(x^16 - x^15 + 25*x^14 - 71*x^13 + 664*x^12 + 2717*x^11 + 14056*x^10 + 38603*x^9 + 219711*x^8 + 318736*x^7 + 672889*x^6 + 1023646*x^5 + 1741879*x^4 - 1221277*x^3 + 992380*x^2 - 536558*x + 707281, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 25 x^{14} - 71 x^{13} + 664 x^{12} + 2717 x^{11} + 14056 x^{10} + 38603 x^{9} + 219711 x^{8} + 318736 x^{7} + 672889 x^{6} + 1023646 x^{5} + 1741879 x^{4} - 1221277 x^{3} + 992380 x^{2} - 536558 x + 707281 \)

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:  \(32790240335000876777587890625=5^{12}\cdot 7^{8}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.57$
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:  $5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(455=5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{455}(64,·)$, $\chi_{455}(1,·)$, $\chi_{455}(398,·)$, $\chi_{455}(337,·)$, $\chi_{455}(274,·)$, $\chi_{455}(83,·)$, $\chi_{455}(216,·)$, $\chi_{455}(92,·)$, $\chi_{455}(34,·)$, $\chi_{455}(356,·)$, $\chi_{455}(428,·)$, $\chi_{455}(174,·)$, $\chi_{455}(307,·)$, $\chi_{455}(246,·)$, $\chi_{455}(183,·)$, $\chi_{455}(447,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{5} - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{6} - \frac{1}{4} a$, $\frac{1}{68} a^{12} - \frac{3}{34} a^{11} - \frac{1}{17} a^{10} + \frac{2}{17} a^{9} + \frac{4}{17} a^{8} + \frac{23}{68} a^{7} + \frac{15}{34} a^{6} - \frac{7}{17} a^{5} + \frac{4}{17} a^{4} + \frac{2}{17} a^{3} - \frac{21}{68} a^{2} - \frac{3}{34} a - \frac{4}{17}$, $\frac{1}{37830400786978628} a^{13} + \frac{104319411994351}{18915200393489314} a^{12} - \frac{53108588345430}{9457600196744657} a^{11} - \frac{4237834067634111}{37830400786978628} a^{10} + \frac{3366138963052668}{9457600196744657} a^{9} - \frac{16630736466900653}{37830400786978628} a^{8} + \frac{7826848711663357}{18915200393489314} a^{7} + \frac{2012658462353323}{9457600196744657} a^{6} + \frac{12026179982069727}{37830400786978628} a^{5} - \frac{614840014634457}{9457600196744657} a^{4} + \frac{6048483208661503}{37830400786978628} a^{3} - \frac{2677666417899667}{18915200393489314} a^{2} - \frac{2934636896332010}{9457600196744657} a - \frac{612256069729409}{1304496578861332}$, $\frac{1}{1097081622822380212} a^{14} - \frac{1}{1097081622822380212} a^{13} + \frac{394473370315432}{274270405705595053} a^{12} - \frac{95417704821144357}{1097081622822380212} a^{11} + \frac{66643574185712911}{548540811411190106} a^{10} - \frac{211231668875808037}{1097081622822380212} a^{9} + \frac{526686522197902553}{1097081622822380212} a^{8} + \frac{15618540503032777}{274270405705595053} a^{7} - \frac{18297061830452679}{1097081622822380212} a^{6} + \frac{222309175227848513}{548540811411190106} a^{5} - \frac{455561820807304145}{1097081622822380212} a^{4} + \frac{391993099671957773}{1097081622822380212} a^{3} + \frac{37499415913494765}{274270405705595053} a^{2} + \frac{12865722513978173}{37830400786978628} a + \frac{80089908241509}{652248289430666}$, $\frac{1}{31815367061849026148} a^{15} - \frac{1}{31815367061849026148} a^{14} + \frac{25}{31815367061849026148} a^{13} + \frac{201847204693919427}{31815367061849026148} a^{12} - \frac{85203690736658191}{15907683530924513074} a^{11} + \frac{1025198318755958221}{15907683530924513074} a^{10} + \frac{5326871234786775345}{31815367061849026148} a^{9} + \frac{2008832694477233599}{31815367061849026148} a^{8} + \frac{10555296811118491285}{31815367061849026148} a^{7} + \frac{2374496301140055505}{15907683530924513074} a^{6} - \frac{2796498281815304717}{7953841765462256537} a^{5} + \frac{9673656437706944281}{31815367061849026148} a^{4} - \frac{12642386200283080953}{31815367061849026148} a^{3} - \frac{377389679378924743}{1097081622822380212} a^{2} + \frac{6165454456154741}{18915200393489314} a + \frac{540557365985433}{1304496578861332}$

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

Class group and class number

$C_{4}\times C_{40}$, which has order $160$ (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:  \( \frac{6329489532}{16133553276799709} a^{15} - \frac{12131521603}{16133553276799709} a^{14} + \frac{312782274373}{32267106553599418} a^{13} - \frac{2343052744033}{64534213107198836} a^{12} + \frac{8847043993353}{32267106553599418} a^{11} + \frac{27775382438799}{32267106553599418} a^{10} + \frac{68124823290377}{16133553276799709} a^{9} + \frac{284007887503067}{32267106553599418} a^{8} + \frac{4259483943268837}{64534213107198836} a^{7} + \frac{894850043597635}{32267106553599418} a^{6} + \frac{1458696794832025}{32267106553599418} a^{5} + \frac{137297704555761}{16133553276799709} a^{4} - \frac{4163221739673}{1112658846675842} a^{3} - \frac{119685975654131439}{64534213107198836} a^{2} + \frac{107073864583}{38367546437098} a + \frac{443591724701}{38367546437098} \) (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:  \( 925356.94221623 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), 4.0.13456625.1, \(\Q(\sqrt{5}, \sqrt{13})\), 4.0.13456625.2, 4.4.2691325.1, 4.4.107653.1, \(\Q(\zeta_{5})\), 4.0.21125.1, 8.0.181080756390625.7, 8.8.7243230255625.1, 8.0.446265625.1

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 R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13Data not computed