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Properties

Label 10.0.14364651494...4375.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,5^{17}\cdot 11^{9}\cdot 41^{8}$
Root discriminant $2604.53$
Ramified primes $5, 11, 41$
Class number $10275100$ (GRH)
Class group $[5, 2055020]$ (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![7057003038899, 2292431710140, 520459612695, 44322002450, 2025226775, 84944046, 836605, -72160, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 72160*x^7 + 836605*x^6 + 84944046*x^5 + 2025226775*x^4 + 44322002450*x^3 + 520459612695*x^2 + 2292431710140*x + 7057003038899)
 
gp: K = bnfinit(x^10 - 72160*x^7 + 836605*x^6 + 84944046*x^5 + 2025226775*x^4 + 44322002450*x^3 + 520459612695*x^2 + 2292431710140*x + 7057003038899, 1)
 

Normalized defining polynomial

\( x^{10} - 72160 x^{7} + 836605 x^{6} + 84944046 x^{5} + 2025226775 x^{4} + 44322002450 x^{3} + 520459612695 x^{2} + 2292431710140 x + 7057003038899 \)

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:  \(-14364651494456106498421478271484375=-\,5^{17}\cdot 11^{9}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2604.53$
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:  $5, 11, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(11275=5^{2}\cdot 11\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{11275}(1,·)$, $\chi_{11275}(2989,·)$, $\chi_{11275}(3659,·)$, $\chi_{11275}(4321,·)$, $\chi_{11275}(4711,·)$, $\chi_{11275}(4856,·)$, $\chi_{11275}(5594,·)$, $\chi_{11275}(9349,·)$, $\chi_{11275}(9979,·)$, $\chi_{11275}(10916,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{246} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{738} a^{6} - \frac{1}{738} a^{5} + \frac{1}{18} a^{4} + \frac{1}{3} a^{3} - \frac{2}{9} a^{2} - \frac{4}{9} a + \frac{1}{18}$, $\frac{1}{12546} a^{7} - \frac{1}{6273} a^{6} - \frac{1}{2091} a^{5} - \frac{13}{306} a^{4} + \frac{49}{153} a^{3} + \frac{25}{153} a^{2} + \frac{1}{102} a - \frac{19}{306}$, $\frac{1}{61312302} a^{8} - \frac{491}{61312302} a^{7} + \frac{16279}{30656151} a^{6} - \frac{26635}{61312302} a^{5} - \frac{119957}{1495422} a^{4} + \frac{16928}{43983} a^{3} + \frac{732325}{1495422} a^{2} + \frac{320387}{747711} a + \frac{188573}{1495422}$, $\frac{1}{5376358847572707335389960521108222} a^{9} - \frac{1953781250360275913911285}{5376358847572707335389960521108222} a^{8} + \frac{897473773045439914554185269}{35139600310932727682287323667374} a^{7} + \frac{1106825450108489197479979772}{2688179423786353667694980260554111} a^{6} - \frac{253885513905739772840433348334}{141483127567702824615525276871269} a^{5} + \frac{365069032285533426235561818064}{21855117266555720875568945207757} a^{4} - \frac{2150232670782033617785509286525}{43710234533111441751137890415514} a^{3} - \frac{28982294047134649181611151123573}{65565351799667162626706835623271} a^{2} + \frac{7944719946519422631002556038920}{21855117266555720875568945207757} a - \frac{13954698636305275967371109711399}{65565351799667162626706835623271}$

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

Class group and class number

$C_{5}\times C_{2055020}$, which has order $10275100$ (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:  \( 1487469.5436229876 \) (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

\(\Q(\sqrt{-55}) \), 5.5.16160924531640625.6

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$5$5.10.17.5$x^{10} - 5 x^{8} + 55$$10$$1$$17$$C_{10}$$[2]_{2}$
$11$11.10.9.9$x^{10} + 297$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$41$41.10.8.3$x^{10} + 943 x^{5} + 242064$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$