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Properties

Label 16.0.24722989956...4209.5
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 59^{8}$
Root discriminant $91.64$
Ramified primes $17, 59$
Class number $696$ (GRH)
Class group $[696]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![289734503, -187885492, 199952265, -82314412, 51842283, -12048508, 5039632, -254052, 275707, 1738, 21844, -786, 1174, -64, 50, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 50*x^14 - 64*x^13 + 1174*x^12 - 786*x^11 + 21844*x^10 + 1738*x^9 + 275707*x^8 - 254052*x^7 + 5039632*x^6 - 12048508*x^5 + 51842283*x^4 - 82314412*x^3 + 199952265*x^2 - 187885492*x + 289734503)
 
gp: K = bnfinit(x^16 - 4*x^15 + 50*x^14 - 64*x^13 + 1174*x^12 - 786*x^11 + 21844*x^10 + 1738*x^9 + 275707*x^8 - 254052*x^7 + 5039632*x^6 - 12048508*x^5 + 51842283*x^4 - 82314412*x^3 + 199952265*x^2 - 187885492*x + 289734503, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 50 x^{14} - 64 x^{13} + 1174 x^{12} - 786 x^{11} + 21844 x^{10} + 1738 x^{9} + 275707 x^{8} - 254052 x^{7} + 5039632 x^{6} - 12048508 x^{5} + 51842283 x^{4} - 82314412 x^{3} + 199952265 x^{2} - 187885492 x + 289734503 \)

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:  \(24722989956581301387479701814209=17^{14}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.64$
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:  $17, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{34} a^{8} - \frac{1}{17} a^{7} + \frac{3}{17} a^{6} + \frac{7}{17} a^{5} - \frac{15}{34} a^{4} - \frac{7}{17} a^{3} - \frac{11}{34} a^{2} - \frac{15}{34} a + \frac{1}{34}$, $\frac{1}{34} a^{9} + \frac{1}{17} a^{7} - \frac{4}{17} a^{6} + \frac{13}{34} a^{5} - \frac{5}{17} a^{4} - \frac{5}{34} a^{3} - \frac{3}{34} a^{2} + \frac{5}{34} a + \frac{1}{17}$, $\frac{1}{34} a^{10} - \frac{2}{17} a^{7} + \frac{1}{34} a^{6} - \frac{2}{17} a^{5} - \frac{9}{34} a^{4} - \frac{9}{34} a^{3} - \frac{7}{34} a^{2} - \frac{1}{17} a - \frac{1}{17}$, $\frac{1}{34} a^{11} - \frac{7}{34} a^{7} - \frac{7}{17} a^{6} + \frac{13}{34} a^{5} - \frac{1}{34} a^{4} + \frac{5}{34} a^{3} - \frac{6}{17} a^{2} + \frac{3}{17} a + \frac{2}{17}$, $\frac{1}{442} a^{12} - \frac{1}{442} a^{10} - \frac{3}{221} a^{9} + \frac{1}{221} a^{8} - \frac{20}{221} a^{7} + \frac{57}{221} a^{6} - \frac{11}{26} a^{5} + \frac{211}{442} a^{4} - \frac{5}{34} a^{3} + \frac{5}{13} a^{2} + \frac{45}{442} a + \frac{67}{442}$, $\frac{1}{442} a^{13} - \frac{1}{442} a^{11} - \frac{3}{221} a^{10} + \frac{1}{221} a^{9} - \frac{1}{442} a^{8} + \frac{18}{221} a^{7} + \frac{47}{442} a^{6} - \frac{127}{442} a^{5} - \frac{8}{17} a^{4} + \frac{33}{221} a^{3} + \frac{29}{221} a^{2} - \frac{38}{221} a + \frac{3}{34}$, $\frac{1}{442} a^{14} - \frac{3}{221} a^{11} + \frac{1}{442} a^{10} + \frac{3}{221} a^{9} - \frac{1}{442} a^{8} + \frac{111}{442} a^{7} + \frac{7}{34} a^{6} + \frac{56}{221} a^{5} - \frac{76}{221} a^{4} + \frac{16}{221} a^{3} + \frac{21}{221} a^{2} - \frac{75}{221} a + \frac{27}{221}$, $\frac{1}{195007616590074081849026211187987897762346} a^{15} - \frac{24235056228197692021341321853020663164}{97503808295037040924513105593993948881173} a^{14} + \frac{211488150713218426677734338029925099119}{195007616590074081849026211187987897762346} a^{13} + \frac{187274274512264112889358198909729995519}{195007616590074081849026211187987897762346} a^{12} - \frac{16212687738261973304077028621082785549}{97503808295037040924513105593993948881173} a^{11} - \frac{774531989266296871874206774949817822767}{97503808295037040924513105593993948881173} a^{10} + \frac{207573051253434592044157163282163859189}{195007616590074081849026211187987897762346} a^{9} + \frac{863089690808770238491901144552135964269}{97503808295037040924513105593993948881173} a^{8} + \frac{88826896640411552356750672062060817299211}{195007616590074081849026211187987897762346} a^{7} + \frac{3148680476346094604789600056099446850863}{7500292945772080071116392737999534529321} a^{6} - \frac{10209186866477901384875771753219048929490}{97503808295037040924513105593993948881173} a^{5} + \frac{44471990545029361661303330954943384326306}{97503808295037040924513105593993948881173} a^{4} - \frac{8599449903791874663738167189897933444166}{97503808295037040924513105593993948881173} a^{3} - \frac{78836502315548426245428373976354569320111}{195007616590074081849026211187987897762346} a^{2} + \frac{34951640994359175595853003229567418830393}{195007616590074081849026211187987897762346} a + \frac{41334875315187904250064305144137378915111}{97503808295037040924513105593993948881173}$

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

Class group and class number

$C_{696}$, which has order $696$ (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:  \( 964643.943008 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

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 $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-59}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-1003}) \), \(\Q(\sqrt{17}, \sqrt{-59})\), 4.4.4913.1, 4.0.17102153.2, 8.0.292483637235409.3, 8.4.1428388920713.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
$59$59.8.4.1$x^{8} + 97468 x^{4} - 205379 x^{2} + 2375002756$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
59.8.4.1$x^{8} + 97468 x^{4} - 205379 x^{2} + 2375002756$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$