↑

Properties

Label 16.0.55005256933...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 37^{14}$
Root discriminant $96.33$
Ramified primes $5, 37$
Class number $338$ (GRH)
Class group $[13, 26]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![130734151, -2254022, 86324549, -39342142, 7304373, 7062268, -2988049, 508734, 226478, -111182, 40339, -9146, 2493, -424, 81, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 81*x^14 - 424*x^13 + 2493*x^12 - 9146*x^11 + 40339*x^10 - 111182*x^9 + 226478*x^8 + 508734*x^7 - 2988049*x^6 + 7062268*x^5 + 7304373*x^4 - 39342142*x^3 + 86324549*x^2 - 2254022*x + 130734151)
 
gp: K = bnfinit(x^16 - 6*x^15 + 81*x^14 - 424*x^13 + 2493*x^12 - 9146*x^11 + 40339*x^10 - 111182*x^9 + 226478*x^8 + 508734*x^7 - 2988049*x^6 + 7062268*x^5 + 7304373*x^4 - 39342142*x^3 + 86324549*x^2 - 2254022*x + 130734151, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 81 x^{14} - 424 x^{13} + 2493 x^{12} - 9146 x^{11} + 40339 x^{10} - 111182 x^{9} + 226478 x^{8} + 508734 x^{7} - 2988049 x^{6} + 7062268 x^{5} + 7304373 x^{4} - 39342142 x^{3} + 86324549 x^{2} - 2254022 x + 130734151 \)

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:  \(55005256933563283079156494140625=5^{14}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.33$
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:  $5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{10} - \frac{1}{4}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{10} + \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{3}{16} a + \frac{7}{16}$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{10} + \frac{3}{16} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{3}{16} a^{5} - \frac{1}{2} a^{4} + \frac{1}{16} a^{3} - \frac{5}{32} a^{2} + \frac{1}{8} a + \frac{1}{32}$, $\frac{1}{128} a^{13} + \frac{1}{128} a^{12} - \frac{3}{128} a^{11} + \frac{13}{128} a^{10} - \frac{9}{64} a^{9} - \frac{3}{64} a^{8} + \frac{1}{8} a^{7} - \frac{11}{64} a^{6} + \frac{3}{64} a^{5} - \frac{27}{64} a^{4} - \frac{27}{128} a^{3} + \frac{15}{128} a^{2} + \frac{41}{128} a + \frac{37}{128}$, $\frac{1}{72704} a^{14} - \frac{51}{18176} a^{13} + \frac{1}{284} a^{12} + \frac{305}{18176} a^{11} + \frac{7461}{72704} a^{10} + \frac{1345}{18176} a^{9} - \frac{8337}{36352} a^{8} + \frac{4557}{36352} a^{7} + \frac{3577}{18176} a^{6} + \frac{439}{18176} a^{5} + \frac{17795}{72704} a^{4} - \frac{6377}{36352} a^{3} + \frac{16555}{36352} a^{2} + \frac{385}{1136} a - \frac{30113}{72704}$, $\frac{1}{74487835063303340724966175978897350410518728876032} a^{15} + \frac{12492540128266695998318878440698862012533107}{74487835063303340724966175978897350410518728876032} a^{14} - \frac{20299697460146803293603902549736670250444705765}{18621958765825835181241543994724337602629682219008} a^{13} + \frac{289088830411621430010046258786798735318116699289}{18621958765825835181241543994724337602629682219008} a^{12} + \frac{1807244744542866963440839470523983609894364245761}{74487835063303340724966175978897350410518728876032} a^{11} - \frac{2925315120994168956727391273465015341738304166465}{74487835063303340724966175978897350410518728876032} a^{10} + \frac{4916167528887737846302041823516456837428302731341}{37243917531651670362483087989448675205259364438016} a^{9} - \frac{4177148956736533687370194794830306595495603916577}{18621958765825835181241543994724337602629682219008} a^{8} + \frac{5208280111316232366711116998275575720025079913829}{37243917531651670362483087989448675205259364438016} a^{7} - \frac{1014992708881202352502673111722352464193494382457}{9310979382912917590620771997362168801314841109504} a^{6} - \frac{1648958600134735410838397348605018752702436939417}{74487835063303340724966175978897350410518728876032} a^{5} + \frac{11193733332840830624455389632872263294892987245739}{74487835063303340724966175978897350410518728876032} a^{4} - \frac{250971138349520387943169267806116493421891139783}{9310979382912917590620771997362168801314841109504} a^{3} + \frac{10667180793515371854410628498009897561556551657189}{37243917531651670362483087989448675205259364438016} a^{2} - \frac{18655735446766748002570656728777607621214736303553}{74487835063303340724966175978897350410518728876032} a + \frac{46350888899973638281103186844238678085271413443}{151706385057644278462252904234006823646677655552}$

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

Class group and class number

$C_{13}\times C_{26}$, which has order $338$ (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:  \( 2969685.17473 \) (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

\(\Q(\sqrt{37}) \), \(\Q(\sqrt{185}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{37})\), 4.4.6331625.1, 4.4.6331625.2, 8.8.40089475140625.1, 8.0.7416552901015625.1 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
5Data not computed
37Data not computed