↑

Properties

Label 8.0.884137376053...3913.1
Degree $8$
Signature $[0, 4]$
Discriminant $97^{7}\cdot 149^{6}$
Root discriminant $2335.15$
Ramified primes $97, 149$
Class number $463033828$ (GRH)
Class group $[463033828]$ (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![374374573056, 52057482048, 3192104472, -20768516, 5308046, -53291, 1801, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 1801*x^6 - 53291*x^5 + 5308046*x^4 - 20768516*x^3 + 3192104472*x^2 + 52057482048*x + 374374573056)
 
gp: K = bnfinit(x^8 - x^7 + 1801*x^6 - 53291*x^5 + 5308046*x^4 - 20768516*x^3 + 3192104472*x^2 + 52057482048*x + 374374573056, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 1801 x^{6} - 53291 x^{5} + 5308046 x^{4} - 20768516 x^{3} + 3192104472 x^{2} + 52057482048 x + 374374573056 \)

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:  \(884137376053502406583563913=97^{7}\cdot 149^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2335.15$
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:  $97, 149$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(14453=97\cdot 149\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $\frac{1}{6} a^{2} + \frac{1}{6} a$, $\frac{1}{6} a^{3} - \frac{1}{6} a$, $\frac{1}{432} a^{4} - \frac{17}{216} a^{3} + \frac{19}{432} a^{2} - \frac{3}{8} a - \frac{1}{3}$, $\frac{1}{1296} a^{5} - \frac{1}{1296} a^{4} + \frac{49}{1296} a^{3} + \frac{35}{432} a^{2} - \frac{5}{72} a + \frac{1}{3}$, $\frac{1}{23328} a^{6} + \frac{1}{7776} a^{5} - \frac{1}{864} a^{4} - \frac{59}{23328} a^{3} + \frac{79}{3888} a^{2} + \frac{7}{324} a - \frac{4}{27}$, $\frac{1}{4738028063483759970816} a^{7} - \frac{101506362472461563}{4738028063483759970816} a^{6} + \frac{164759677724305069}{1579342687827919990272} a^{5} - \frac{1909037976750310721}{4738028063483759970816} a^{4} - \frac{32184908683801214875}{592253507935469996352} a^{3} - \frac{132274791735625351}{394835671956979997568} a^{2} + \frac{4180376165257693069}{16451486331540833232} a - \frac{9512038001338481}{31158118052160669}$

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

Class group and class number

$C_{463033828}$, which has order $463033828$ (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:  \( 79274.2426916 \) (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

\(\Q(\sqrt{97}) \), 4.4.20262253273.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$97$97.8.7.4$x^{8} - 1515625$$8$$1$$7$$C_8$$[\ ]_{8}$
$149$149.8.6.3$x^{8} - 149 x^{4} + 66603$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$