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Properties

Label 16.0.26268600651...000.63
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{12}$
Root discriminant $387.90$
Ramified primes $2, 3, 5, 17$
Class number $201932800$ (GRH)
Class group $[2, 2, 8, 8, 40, 19720]$ (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![244628170801, 0, 57170356256, 0, 10689323324, 0, 1108555112, 0, 64573714, 0, 2154304, 0, 39116, 0, 328, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 328*x^14 + 39116*x^12 + 2154304*x^10 + 64573714*x^8 + 1108555112*x^6 + 10689323324*x^4 + 57170356256*x^2 + 244628170801)
 
gp: K = bnfinit(x^16 + 328*x^14 + 39116*x^12 + 2154304*x^10 + 64573714*x^8 + 1108555112*x^6 + 10689323324*x^4 + 57170356256*x^2 + 244628170801, 1)
 

Normalized defining polynomial

\( x^{16} + 328 x^{14} + 39116 x^{12} + 2154304 x^{10} + 64573714 x^{8} + 1108555112 x^{6} + 10689323324 x^{4} + 57170356256 x^{2} + 244628170801 \)

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:  \(262686006513622818702917369856000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $387.90$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4080=2^{4}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{4080}(1,·)$, $\chi_{4080}(2053,·)$, $\chi_{4080}(2891,·)$, $\chi_{4080}(13,·)$, $\chi_{4080}(851,·)$, $\chi_{4080}(2197,·)$, $\chi_{4080}(2903,·)$, $\chi_{4080}(157,·)$, $\chi_{4080}(863,·)$, $\chi_{4080}(2209,·)$, $\chi_{4080}(3047,·)$, $\chi_{4080}(169,·)$, $\chi_{4080}(1007,·)$, $\chi_{4080}(3059,·)$, $\chi_{4080}(2041,·)$, $\chi_{4080}(1019,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{84} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{2}{21} a^{2} - \frac{5}{12}$, $\frac{1}{84} a^{9} - \frac{1}{21} a^{7} - \frac{1}{3} a^{5} - \frac{2}{21} a^{3} + \frac{25}{84} a$, $\frac{1}{84} a^{10} + \frac{1}{3} a^{6} - \frac{3}{7} a^{4} - \frac{1}{12} a^{2} + \frac{1}{3}$, $\frac{1}{84} a^{11} + \frac{1}{21} a^{7} - \frac{3}{7} a^{5} - \frac{1}{12} a^{3} - \frac{8}{21} a$, $\frac{1}{295092} a^{12} - \frac{93}{32788} a^{10} - \frac{271}{49182} a^{8} - \frac{2606}{73773} a^{6} - \frac{6907}{32788} a^{4} - \frac{24035}{98364} a^{2} + \frac{3403}{21078}$, $\frac{1}{20850315444} a^{13} - \frac{21157907}{6950105148} a^{11} + \frac{1994663}{1737526287} a^{9} + \frac{112480141}{5212578861} a^{7} + \frac{2579781275}{6950105148} a^{5} - \frac{3022384403}{6950105148} a^{3} + \frac{693954122}{5212578861} a$, $\frac{1}{458740177353641575847656908} a^{14} - \frac{8118115896095105891}{7281590116724469457899316} a^{12} + \frac{19346392429674979516991}{3640795058362234728949658} a^{10} + \frac{1960274046650992986217771}{458740177353641575847656908} a^{8} - \frac{10080136527427269584271587}{21844770350173408373697948} a^{6} + \frac{3237548556631320449688605}{7281590116724469457899316} a^{4} + \frac{61962310384155367416956359}{229370088676820787923828454} a^{2} - \frac{3725722552253286253}{14722209540909847084}$, $\frac{1}{458740177353641575847656908} a^{15} + \frac{1107871948125827}{65534311050520225121093844} a^{13} + \frac{760572030510969854166}{1820397529181117364474829} a^{11} - \frac{11393220635917322774651}{458740177353641575847656908} a^{9} - \frac{2348974139279887313011717}{65534311050520225121093844} a^{7} - \frac{1362699754640078886947203}{21844770350173408373697948} a^{5} + \frac{57200540901869155676742446}{114685044338410393961914227} a^{3} - \frac{9197386006814462581843025}{65534311050520225121093844} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{8}\times C_{8}\times C_{40}\times C_{19720}$, which has order $201932800$ (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:  \( -1 \) (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:  \( 6814399.346097333 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{170}) \), \(\Q(\sqrt{85}) \), 4.4.18432.1, \(\Q(\sqrt{2}, \sqrt{85})\), 4.4.133171200.2, 4.0.1257728000.2, 4.0.1257728000.8, 4.0.353736000.4, 4.0.88434000.2, 8.8.17734568509440000.21, 8.0.1581879721984000000.4, 8.0.2002066523136000000.24

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 R R R ${\href{/LocalNumberField/7.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$17$17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$