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Properties

Label 10.2.16042489871...8125.1
Degree $10$
Signature $[2, 4]$
Discriminant $3^{8}\cdot 5^{11}\cdot 7^{8}\cdot 233^{8}$
Root discriminant $5254.45$
Ramified primes $3, 5, 7, 233$
Class number $7805722500$ (GRH)
Class group $[5, 5, 17670, 17670]$ (GRH)
Galois group $F_5$ (as 10T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2053364713394525, 1585992550, -1812562925, 1359422200, -226570350, 90628135, -15, 10, 5, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 5*x^9 + 5*x^8 + 10*x^7 - 15*x^6 + 90628135*x^5 - 226570350*x^4 + 1359422200*x^3 - 1812562925*x^2 + 1585992550*x + 2053364713394525)
 
gp: K = bnfinit(x^10 - 5*x^9 + 5*x^8 + 10*x^7 - 15*x^6 + 90628135*x^5 - 226570350*x^4 + 1359422200*x^3 - 1812562925*x^2 + 1585992550*x + 2053364713394525, 1)
 

Normalized defining polynomial

\( x^{10} - 5 x^{9} + 5 x^{8} + 10 x^{7} - 15 x^{6} + 90628135 x^{5} - 226570350 x^{4} + 1359422200 x^{3} - 1812562925 x^{2} + 1585992550 x + 2053364713394525 \)

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

Invariants

Degree:  $10$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(16042489871729878088387919882861328125=3^{8}\cdot 5^{11}\cdot 7^{8}\cdot 233^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $5254.45$
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:  $3, 5, 7, 233$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{21} a^{4} - \frac{2}{21} a^{3} - \frac{1}{21} a^{2} + \frac{2}{21} a + \frac{1}{21}$, $\frac{1}{105} a^{5} - \frac{1}{21} a^{3} + \frac{1}{21} a - \frac{8}{21}$, $\frac{1}{2205} a^{6} - \frac{1}{735} a^{5} + \frac{1}{441} a^{3} - \frac{59}{147} a + \frac{88}{441}$, $\frac{1}{2205} a^{7} - \frac{1}{245} a^{5} + \frac{1}{441} a^{4} + \frac{1}{147} a^{3} - \frac{59}{147} a^{2} - \frac{2}{441} a - \frac{59}{147}$, $\frac{1}{231525} a^{8} - \frac{1}{9261} a^{7} - \frac{8}{46305} a^{6} - \frac{40}{9261} a^{5} - \frac{904}{46305} a^{4} - \frac{724}{46305} a^{3} + \frac{179}{9261} a^{2} - \frac{2456}{9261} a - \frac{278}{9261}$, $\frac{1}{976180870201076659794376369783516275} a^{9} - \frac{594882826570168622080318378633}{976180870201076659794376369783516275} a^{8} - \frac{5945148577749445179520478892026}{195236174040215331958875273956703255} a^{7} + \frac{26942979151347912774766106475004}{195236174040215331958875273956703255} a^{6} + \frac{42015592788224554658417729022176}{27890882005745047422696467708100465} a^{5} - \frac{1128077223882746090372286718637662}{195236174040215331958875273956703255} a^{4} + \frac{88105408959717544542728769195344272}{195236174040215331958875273956703255} a^{3} + \frac{657035500882305230858181794289544}{5578176401149009484539293541620093} a^{2} + \frac{2699900151874108967415710199472171}{5578176401149009484539293541620093} a - \frac{1167808221304519320265298962670462}{13015744936014355463925018263780217}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{17670}\times C_{17670}$, which has order $7805722500$ (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:  $5$
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:  \( 68008.13611618712 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 10T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.1791228063186253125.1

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

Sibling fields

Galois closure: data not computed
Degree 5 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$7$7.10.8.1$x^{10} - 7 x^{5} + 147$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
233Data not computed