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Properties

Label 16.8.83120476897...3681.1
Degree $16$
Signature $[8, 4]$
Discriminant $23^{12}\cdot 41^{14}$
Root discriminant $270.70$
Ramified primes $23, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4615661, 66208918, -63183866, -64346268, 22083648, 35482368, 10582866, -2155488, -3549491, -1284806, -141016, -21718, -7707, -98, 33, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 33*x^14 - 98*x^13 - 7707*x^12 - 21718*x^11 - 141016*x^10 - 1284806*x^9 - 3549491*x^8 - 2155488*x^7 + 10582866*x^6 + 35482368*x^5 + 22083648*x^4 - 64346268*x^3 - 63183866*x^2 + 66208918*x - 4615661)
 
gp: K = bnfinit(x^16 - 4*x^15 + 33*x^14 - 98*x^13 - 7707*x^12 - 21718*x^11 - 141016*x^10 - 1284806*x^9 - 3549491*x^8 - 2155488*x^7 + 10582866*x^6 + 35482368*x^5 + 22083648*x^4 - 64346268*x^3 - 63183866*x^2 + 66208918*x - 4615661, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 33 x^{14} - 98 x^{13} - 7707 x^{12} - 21718 x^{11} - 141016 x^{10} - 1284806 x^{9} - 3549491 x^{8} - 2155488 x^{7} + 10582866 x^{6} + 35482368 x^{5} + 22083648 x^{4} - 64346268 x^{3} - 63183866 x^{2} + 66208918 x - 4615661 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(831204768973346090928597984067970713681=23^{12}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $270.70$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{10} a^{8} + \frac{1}{10} a^{7} - \frac{1}{10} a^{6} + \frac{3}{10} a^{5} + \frac{2}{5} a^{4} + \frac{3}{10} a^{3} + \frac{1}{10} a - \frac{3}{10}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{11} + \frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{1}{5} a^{8} - \frac{3}{10} a^{7} - \frac{1}{2} a^{6} + \frac{3}{10} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{10}$, $\frac{1}{10} a^{14} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{3}{10} a^{7} + \frac{1}{10} a^{6} + \frac{1}{10} a^{5} + \frac{1}{5} a^{4} - \frac{3}{10} a^{3} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{228190432720689356929193462561358567942500047284495171366255490} a^{15} + \frac{4140814808448996092782445514662610267019534865453327978919042}{114095216360344678464596731280679283971250023642247585683127745} a^{14} - \frac{5444013697522628544960852971249554321789587479313675042279188}{114095216360344678464596731280679283971250023642247585683127745} a^{13} + \frac{3436132640714749317048150504747617513687045239390063465362641}{228190432720689356929193462561358567942500047284495171366255490} a^{12} - \frac{14458865649765637855211871138261031341681074552337481319793174}{114095216360344678464596731280679283971250023642247585683127745} a^{11} - \frac{37782275510438094163198193712066315691548330992336405990435849}{228190432720689356929193462561358567942500047284495171366255490} a^{10} + \frac{33015752323349824493498232343924196361175050417683661258278333}{228190432720689356929193462561358567942500047284495171366255490} a^{9} - \frac{11179694377320922363786762152844395470198133568822764814808451}{114095216360344678464596731280679283971250023642247585683127745} a^{8} - \frac{14065973172775301453028885360898868928058362810888668384294087}{228190432720689356929193462561358567942500047284495171366255490} a^{7} - \frac{10498189731427598916684558046767638632813890149225283936637253}{114095216360344678464596731280679283971250023642247585683127745} a^{6} - \frac{50996815171192196897270096183513877691240294418160217280267099}{228190432720689356929193462561358567942500047284495171366255490} a^{5} - \frac{42009824258841062881736360745315574993990483363518186476556203}{114095216360344678464596731280679283971250023642247585683127745} a^{4} + \frac{30234248275466037053045641411737324051770005972463796570264787}{228190432720689356929193462561358567942500047284495171366255490} a^{3} + \frac{44912175367008945209662010531381494782529771635374287663736937}{228190432720689356929193462561358567942500047284495171366255490} a^{2} + \frac{78853967716501672754684871562015032341094215448526907711748567}{228190432720689356929193462561358567942500047284495171366255490} a + \frac{4820799914800974866659320982449010616125740997360308115124656}{114095216360344678464596731280679283971250023642247585683127745}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 8117451696600 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.54500230757132921.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$23$23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$