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Properties

Label 10.0.81500110851...0000.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,2^{15}\cdot 5^{5}\cdot 11^{8}\cdot 13^{5}$
Root discriminant $155.28$
Ramified primes $2, 5, 11, 13$
Class number $178244$ (GRH)
Class group $[2, 89122]$ (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![39761899451, -593461186, 1472396383, -17679454, 22072947, -200492, 167456, -1026, 643, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 + 643*x^8 - 1026*x^7 + 167456*x^6 - 200492*x^5 + 22072947*x^4 - 17679454*x^3 + 1472396383*x^2 - 593461186*x + 39761899451)
 
gp: K = bnfinit(x^10 - 2*x^9 + 643*x^8 - 1026*x^7 + 167456*x^6 - 200492*x^5 + 22072947*x^4 - 17679454*x^3 + 1472396383*x^2 - 593461186*x + 39761899451, 1)
 

Normalized defining polynomial

\( x^{10} - 2 x^{9} + 643 x^{8} - 1026 x^{7} + 167456 x^{6} - 200492 x^{5} + 22072947 x^{4} - 17679454 x^{3} + 1472396383 x^{2} - 593461186 x + 39761899451 \)

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:  \(-8150011085120819200000=-\,2^{15}\cdot 5^{5}\cdot 11^{8}\cdot 13^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $155.28$
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:  $2, 5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(5720=2^{3}\cdot 5\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{5720}(1,·)$, $\chi_{5720}(1819,·)$, $\chi_{5720}(4161,·)$, $\chi_{5720}(1299,·)$, $\chi_{5720}(521,·)$, $\chi_{5720}(779,·)$, $\chi_{5720}(5201,·)$, $\chi_{5720}(5459,·)$, $\chi_{5720}(2601,·)$, $\chi_{5720}(3899,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{1572180156064001531590605481} a^{9} - \frac{405715607132855991566607977}{1572180156064001531590605481} a^{8} - \frac{697591187651372541482458815}{1572180156064001531590605481} a^{7} + \frac{609731791712175468002209192}{1572180156064001531590605481} a^{6} - \frac{301697931456935964758390514}{1572180156064001531590605481} a^{5} + \frac{637457405323826305991268309}{1572180156064001531590605481} a^{4} + \frac{669892407637603594723954337}{1572180156064001531590605481} a^{3} - \frac{709103951898774267413158493}{1572180156064001531590605481} a^{2} + \frac{377269014068659572518399268}{1572180156064001531590605481} a - \frac{11803584792291664220796918}{68355658959304414416982847}$

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

Class group and class number

$C_{2}\times C_{89122}$, which has order $178244$ (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:  \( 26.1711060094 \) (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

\(\Q(\sqrt{-130}) \), \(\Q(\zeta_{11})^+\)

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }$ R ${\href{/LocalNumberField/7.10.0.1}{10} }$ R R ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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
$2$2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$13$13.10.5.1$x^{10} - 676 x^{6} + 114244 x^{2} - 13366548$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$