Properties

Label 16.16.685...361.1
Degree $16$
Signature $[16, 0]$
Discriminant $6.856\times 10^{44}$
Root discriminant $634.24$
Ramified primes $13, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 486*x^14 + 3640*x^13 + 85482*x^12 - 680369*x^11 - 6731288*x^10 + 57906718*x^9 + 234236862*x^8 - 2308745163*x^7 - 3167386332*x^6 + 41364170537*x^5 + 15994510202*x^4 - 316660788899*x^3 - 107062029196*x^2 + 853361887750*x + 670030778699)
 
gp: K = bnfinit(x^16 - 7*x^15 - 486*x^14 + 3640*x^13 + 85482*x^12 - 680369*x^11 - 6731288*x^10 + 57906718*x^9 + 234236862*x^8 - 2308745163*x^7 - 3167386332*x^6 + 41364170537*x^5 + 15994510202*x^4 - 316660788899*x^3 - 107062029196*x^2 + 853361887750*x + 670030778699, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![670030778699, 853361887750, -107062029196, -316660788899, 15994510202, 41364170537, -3167386332, -2308745163, 234236862, 57906718, -6731288, -680369, 85482, 3640, -486, -7, 1]);
 

\( x^{16} - 7 x^{15} - 486 x^{14} + 3640 x^{13} + 85482 x^{12} - 680369 x^{11} - 6731288 x^{10} + 57906718 x^{9} + 234236862 x^{8} - 2308745163 x^{7} - 3167386332 x^{6} + 41364170537 x^{5} + 15994510202 x^{4} - 316660788899 x^{3} - 107062029196 x^{2} + 853361887750 x + 670030778699 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[16, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(685578251337344185848967768724165145970351361\)\(\medspace = 13^{14}\cdot 89^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $634.24$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $13, 89$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{12049} a^{14} - \frac{5289}{12049} a^{13} - \frac{756}{12049} a^{12} + \frac{772}{12049} a^{11} - \frac{5486}{12049} a^{10} + \frac{579}{12049} a^{9} + \frac{4339}{12049} a^{8} - \frac{5965}{12049} a^{7} - \frac{5994}{12049} a^{6} + \frac{1607}{12049} a^{5} - \frac{1377}{12049} a^{4} + \frac{4540}{12049} a^{3} + \frac{5030}{12049} a^{2} - \frac{3946}{12049} a - \frac{1677}{12049}$, $\frac{1}{72057101988746016533572780707625737993804032276052729776719294788811893089} a^{15} + \frac{463088400443828630663301795945766765370165013181884508284542312365887}{72057101988746016533572780707625737993804032276052729776719294788811893089} a^{14} + \frac{17455327169192467030077843450879448071820107396686706507544392616119164718}{72057101988746016533572780707625737993804032276052729776719294788811893089} a^{13} - \frac{19422213533814682133077739065496257987629072822179776996861237695753271820}{72057101988746016533572780707625737993804032276052729776719294788811893089} a^{12} - \frac{22485600772272588602766051603604589032442611302200404796178380107549174140}{72057101988746016533572780707625737993804032276052729776719294788811893089} a^{11} - \frac{23738103892741485110225204151910293236262647545351585887036388980524471187}{72057101988746016533572780707625737993804032276052729776719294788811893089} a^{10} - \frac{1023422726911297613704115403424378166897092196264943077339285215847560573}{72057101988746016533572780707625737993804032276052729776719294788811893089} a^{9} + \frac{28724875867031420501638552323676487287025868066605872425855418864594118947}{72057101988746016533572780707625737993804032276052729776719294788811893089} a^{8} - \frac{10483787038636549151671906733315718344400425957274709445683653728581364930}{72057101988746016533572780707625737993804032276052729776719294788811893089} a^{7} + \frac{12256881219381711072383534542283599543297878201026267663594531223078605744}{72057101988746016533572780707625737993804032276052729776719294788811893089} a^{6} + \frac{15554303298670618573515435495027991062111731524871092275512473021049315137}{72057101988746016533572780707625737993804032276052729776719294788811893089} a^{5} + \frac{4430845720692003916462822445651403314445640739708108996830581417120097998}{72057101988746016533572780707625737993804032276052729776719294788811893089} a^{4} + \frac{271612133996991761818436694627269209847632498608487694104442818910658546}{72057101988746016533572780707625737993804032276052729776719294788811893089} a^{3} - \frac{4839691048212396700940084101421293303892015499220050379334516669160116193}{72057101988746016533572780707625737993804032276052729776719294788811893089} a^{2} - \frac{6253752061529753133119134588665920483919651665135459843311275049077981908}{72057101988746016533572780707625737993804032276052729776719294788811893089} a + \frac{15818019925732466468457867744240110233149815839406002503092758127088269163}{72057101988746016533572780707625737993804032276052729776719294788811893089}$

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

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 55517458750200000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{16}\cdot(2\pi)^{0}\cdot 55517458750200000 \cdot 2}{2\sqrt{685578251337344185848967768724165145970351361}}\approx 0.138957180399297$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.119139761.1, 8.8.213496205355753436961.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
13Data not computed
89Data not computed