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Properties

Label 10.0.34979630064...1875.2
Degree $10$
Signature $[0, 5]$
Discriminant $-\,5^{17}\cdot 71^{9}$
Root discriminant $715.12$
Ramified primes $5, 71$
Class number $1310720$ (GRH)
Class group $[4, 16, 16, 16, 80]$ (GRH)
Galois group $C_{10}$ (as 10T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3628410199, 3428920860, 1053034370, 135006500, 13287650, 296354, -4970, -4615, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 4615*x^7 - 4970*x^6 + 296354*x^5 + 13287650*x^4 + 135006500*x^3 + 1053034370*x^2 + 3428920860*x + 3628410199)
 
gp: K = bnfinit(x^10 - 4615*x^7 - 4970*x^6 + 296354*x^5 + 13287650*x^4 + 135006500*x^3 + 1053034370*x^2 + 3428920860*x + 3628410199, 1)
 

Normalized defining polynomial

\( x^{10} - 4615 x^{7} - 4970 x^{6} + 296354 x^{5} + 13287650 x^{4} + 135006500 x^{3} + 1053034370 x^{2} + 3428920860 x + 3628410199 \)

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

Invariants

Degree:  $10$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-34979630064734673309326171875=-\,5^{17}\cdot 71^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $715.12$
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:  $5, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1775=5^{2}\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{1775}(1,·)$, $\chi_{1775}(196,·)$, $\chi_{1775}(806,·)$, $\chi_{1775}(1761,·)$, $\chi_{1775}(969,·)$, $\chi_{1775}(1579,·)$, $\chi_{1775}(14,·)$, $\chi_{1775}(1774,·)$, $\chi_{1775}(634,·)$, $\chi_{1775}(1141,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{630713415306086363566436398688270921} a^{9} + \frac{150771466646953480298292962087242879}{630713415306086363566436398688270921} a^{8} + \frac{55009991432788797734436422461463769}{630713415306086363566436398688270921} a^{7} + \frac{277493248345594034226887780747060653}{630713415306086363566436398688270921} a^{6} + \frac{84371041428569610011496484864004380}{630713415306086363566436398688270921} a^{5} - \frac{295693486812452418470658044820226444}{630713415306086363566436398688270921} a^{4} - \frac{283954367693788250662528520882010303}{630713415306086363566436398688270921} a^{3} - \frac{53827242563760262605775598503896343}{630713415306086363566436398688270921} a^{2} - \frac{50746652618371703556321869353132088}{630713415306086363566436398688270921} a - \frac{240134277553176076707204961596360623}{630713415306086363566436398688270921}$

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

Class group and class number

$C_{4}\times C_{16}\times C_{16}\times C_{16}\times C_{80}$, which has order $1310720$ (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:  $4$
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:  \( 13426.791976539287 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}$ (as 10T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 10
The 10 conjugacy class representatives for $C_{10}$
Character table for $C_{10}$

Intermediate fields

\(\Q(\sqrt{-355}) \), 5.5.9926437890625.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }$ ${\href{/LocalNumberField/3.10.0.1}{10} }$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$5$5.10.17.5$x^{10} - 5 x^{8} + 55$$10$$1$$17$$C_{10}$$[2]_{2}$
$71$71.10.9.6$x^{10} + 142$$10$$1$$9$$C_{10}$$[\ ]_{10}$