↑

Properties

Label 8.0.434582889001...9241.1
Degree $8$
Signature $[0, 4]$
Discriminant $281^{7}\cdot 421^{4}$
Root discriminant $2849.44$
Ramified primes $281, 421$
Class number $317215232$ (GRH)
Class group $[4, 4, 4, 4956488]$ (GRH)
Galois group $C_8$ (as 8T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![354721068818432, 431596924160, 480109130864, -175461772, 207419356, -59155, 29523, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 29523*x^6 - 59155*x^5 + 207419356*x^4 - 175461772*x^3 + 480109130864*x^2 + 431596924160*x + 354721068818432)
 
gp: K = bnfinit(x^8 - x^7 + 29523*x^6 - 59155*x^5 + 207419356*x^4 - 175461772*x^3 + 480109130864*x^2 + 431596924160*x + 354721068818432, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 29523 x^{6} - 59155 x^{5} + 207419356 x^{4} - 175461772 x^{3} + 480109130864 x^{2} + 431596924160 x + 354721068818432 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4345828890015370938034129241=281^{7}\cdot 421^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2849.44$
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:  $281, 421$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(118301=281\cdot 421\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{5} - \frac{1}{16} a^{4} - \frac{1}{32} a^{3} + \frac{1}{16} a^{2}$, $\frac{1}{320} a^{6} + \frac{1}{160} a^{5} - \frac{9}{320} a^{4} - \frac{9}{160} a^{3} + \frac{3}{40} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{98944585743915532985426112099754240} a^{7} + \frac{15992164676465006556060298789247}{98944585743915532985426112099754240} a^{6} + \frac{559852952736058684429790808296171}{98944585743915532985426112099754240} a^{5} + \frac{1999051612774692850423171798651677}{98944585743915532985426112099754240} a^{4} - \frac{1165465591552872958444330428166719}{24736146435978883246356528024938560} a^{3} + \frac{5825288046178305657592392984172401}{24736146435978883246356528024938560} a^{2} - \frac{2864003779402755946951543846567061}{6184036608994720811589132006234640} a - \frac{28623451744334452151656724804094}{77300457612434010144864150077933}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{4956488}$, which has order $317215232$ (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:  $3$
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:  \( 15221.0688326 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8$ (as 8T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 8
The 8 conjugacy class representatives for $C_8$
Character table for $C_8$

Intermediate fields

\(\Q(\sqrt{281}) \), 4.4.22188041.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
281Data not computed
421Data not computed