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Properties

Label 16.16.1048315135...4369.2
Degree $16$
Signature $[16, 0]$
Discriminant $17^{14}\cdot 53^{8}$
Root discriminant $86.85$
Ramified primes $17, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![30259, 54341, -165824, -220035, 315322, 304908, -241469, -187170, 76870, 49215, -12476, -5904, 1129, 316, -54, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 54*x^14 + 316*x^13 + 1129*x^12 - 5904*x^11 - 12476*x^10 + 49215*x^9 + 76870*x^8 - 187170*x^7 - 241469*x^6 + 304908*x^5 + 315322*x^4 - 220035*x^3 - 165824*x^2 + 54341*x + 30259)
 
gp: K = bnfinit(x^16 - 6*x^15 - 54*x^14 + 316*x^13 + 1129*x^12 - 5904*x^11 - 12476*x^10 + 49215*x^9 + 76870*x^8 - 187170*x^7 - 241469*x^6 + 304908*x^5 + 315322*x^4 - 220035*x^3 - 165824*x^2 + 54341*x + 30259, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 54 x^{14} + 316 x^{13} + 1129 x^{12} - 5904 x^{11} - 12476 x^{10} + 49215 x^{9} + 76870 x^{8} - 187170 x^{7} - 241469 x^{6} + 304908 x^{5} + 315322 x^{4} - 220035 x^{3} - 165824 x^{2} + 54341 x + 30259 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10483151353726139536553735554369=17^{14}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.85$
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:  $17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{26} a^{14} + \frac{2}{13} a^{13} - \frac{3}{13} a^{12} - \frac{11}{26} a^{11} - \frac{2}{13} a^{10} + \frac{11}{26} a^{8} - \frac{5}{13} a^{7} + \frac{1}{13} a^{6} + \frac{9}{26} a^{5} + \frac{4}{13} a^{4} + \frac{1}{13} a^{3} - \frac{1}{2} a^{2} + \frac{3}{13} a + \frac{6}{13}$, $\frac{1}{593268383775801003071303881423018} a^{15} + \frac{2490202065410282607946086500935}{296634191887900501535651940711509} a^{14} - \frac{32375678680999134802243991323891}{593268383775801003071303881423018} a^{13} + \frac{13292512273160109192758174756277}{593268383775801003071303881423018} a^{12} + \frac{76937574525306480214883339204821}{296634191887900501535651940711509} a^{11} - \frac{222082352379546088218273871031549}{593268383775801003071303881423018} a^{10} - \frac{283749633662838353131107555072909}{593268383775801003071303881423018} a^{9} - \frac{115505478672522288051052892248731}{296634191887900501535651940711509} a^{8} + \frac{121208052069177740805463583539805}{593268383775801003071303881423018} a^{7} - \frac{37537474353320669841198175132049}{593268383775801003071303881423018} a^{6} + \frac{127716953854287561993408654788190}{296634191887900501535651940711509} a^{5} - \frac{234435972895186404132473544483931}{593268383775801003071303881423018} a^{4} - \frac{125936009152099776373915537716561}{593268383775801003071303881423018} a^{3} - \frac{5139860005163743577193979318735}{296634191887900501535651940711509} a^{2} - \frac{234840842846585477801917370603731}{593268383775801003071303881423018} a + \frac{76277088807240246067172562606169}{296634191887900501535651940711509}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 7359535457.67 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.4.15317.1, 4.4.260389.1, 8.8.3237769502871713.1, 8.8.1152641332457.1, 8.8.67802431321.1

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 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
$53$53.4.2.2$x^{4} - 53 x^{2} + 14045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
53.4.2.2$x^{4} - 53 x^{2} + 14045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$