↑

Properties

Label 16.0.45641382753...0000.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}$
Root discriminant $71.40$
Ramified primes $2, 3, 5, 19$
Class number $10240$ (GRH)
Class group $[2, 4, 16, 80]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14611195, -7262120, 9158010, -4102400, 2866466, -1090292, 561050, -192320, 78891, -23980, 8426, -2240, 616, -140, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 40*x^14 - 140*x^13 + 616*x^12 - 2240*x^11 + 8426*x^10 - 23980*x^9 + 78891*x^8 - 192320*x^7 + 561050*x^6 - 1090292*x^5 + 2866466*x^4 - 4102400*x^3 + 9158010*x^2 - 7262120*x + 14611195)
 
gp: K = bnfinit(x^16 - 8*x^15 + 40*x^14 - 140*x^13 + 616*x^12 - 2240*x^11 + 8426*x^10 - 23980*x^9 + 78891*x^8 - 192320*x^7 + 561050*x^6 - 1090292*x^5 + 2866466*x^4 - 4102400*x^3 + 9158010*x^2 - 7262120*x + 14611195, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 40 x^{14} - 140 x^{13} + 616 x^{12} - 2240 x^{11} + 8426 x^{10} - 23980 x^{9} + 78891 x^{8} - 192320 x^{7} + 561050 x^{6} - 1090292 x^{5} + 2866466 x^{4} - 4102400 x^{3} + 9158010 x^{2} - 7262120 x + 14611195 \)

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:  \(456413827530756096000000000000=2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.40$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2280=2^{3}\cdot 3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2280}(1,·)$, $\chi_{2280}(1027,·)$, $\chi_{2280}(1217,·)$, $\chi_{2280}(1673,·)$, $\chi_{2280}(1483,·)$, $\chi_{2280}(1937,·)$, $\chi_{2280}(2051,·)$, $\chi_{2280}(1369,·)$, $\chi_{2280}(2203,·)$, $\chi_{2280}(419,·)$, $\chi_{2280}(721,·)$, $\chi_{2280}(1331,·)$, $\chi_{2280}(113,·)$, $\chi_{2280}(1139,·)$, $\chi_{2280}(1747,·)$, $\chi_{2280}(2089,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{8} a^{2} - \frac{1}{8}$, $\frac{1}{248} a^{13} + \frac{9}{248} a^{12} + \frac{5}{248} a^{11} + \frac{13}{248} a^{10} + \frac{5}{62} a^{9} - \frac{1}{31} a^{8} + \frac{37}{248} a^{7} - \frac{39}{248} a^{6} + \frac{7}{62} a^{5} + \frac{5}{62} a^{4} - \frac{51}{248} a^{3} - \frac{71}{248} a^{2} - \frac{109}{248} a + \frac{119}{248}$, $\frac{1}{521010724946003912} a^{14} - \frac{7}{521010724946003912} a^{13} - \frac{2959570629918355}{65126340618250489} a^{12} + \frac{11806708999580153}{521010724946003912} a^{11} - \frac{28774056894441121}{521010724946003912} a^{10} + \frac{6966669218318603}{260505362473001956} a^{9} - \frac{14818069260669}{157929895406488} a^{8} - \frac{18692352133655099}{521010724946003912} a^{7} + \frac{69346564399251451}{521010724946003912} a^{6} + \frac{3729659905292219}{260505362473001956} a^{5} + \frac{39997220924398597}{521010724946003912} a^{4} - \frac{120581723476785951}{521010724946003912} a^{3} + \frac{126242842619230741}{260505362473001956} a^{2} - \frac{154419329773737279}{521010724946003912} a - \frac{125135583386059711}{521010724946003912}$, $\frac{1}{15458909219872882072952} a^{15} + \frac{3707}{3864727304968220518238} a^{14} - \frac{9191784644331229201}{15458909219872882072952} a^{13} - \frac{291507080453077729915}{15458909219872882072952} a^{12} - \frac{107042353170148083059}{1932363652484110259119} a^{11} - \frac{241764342352924228929}{15458909219872882072952} a^{10} + \frac{1850995652421768156955}{15458909219872882072952} a^{9} + \frac{84290823878860514283}{7729454609936441036476} a^{8} - \frac{718921113079819852679}{3864727304968220518238} a^{7} + \frac{3034797891004241599715}{15458909219872882072952} a^{6} - \frac{1071062561750855207877}{15458909219872882072952} a^{5} - \frac{1110027928871025208287}{7729454609936441036476} a^{4} - \frac{252020115141036890841}{15458909219872882072952} a^{3} - \frac{4055690090624029839593}{15458909219872882072952} a^{2} + \frac{591083351827137558466}{1932363652484110259119} a - \frac{6608403352482250444735}{15458909219872882072952}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{16}\times C_{80}$, which has order $10240$ (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:  \( 19771.344992359085 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-95}) \), \(\Q(\sqrt{-570}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-114}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{6}, \sqrt{-95})\), \(\Q(\sqrt{5}, \sqrt{-19})\), \(\Q(\sqrt{30}, \sqrt{-95})\), \(\Q(\sqrt{5}, \sqrt{-114})\), \(\Q(\sqrt{-19}, \sqrt{30})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-19})\), 4.4.8000.1, 4.0.2888000.1, 4.0.406125.2, \(\Q(\zeta_{15})^+\), 8.0.27023362560000.101, 8.0.8340544000000.3, 8.0.164937515625.1, 8.0.675584064000000.187, 8.0.675584064000000.194, 8.8.5184000000.2, 8.0.675584064000000.234

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$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}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$