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Properties

Label 10.0.63465642057...1875.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,3^{5}\cdot 5^{16}\cdot 11^{8}\cdot 41^{8}$
Root discriminant $3021.72$
Ramified primes $3, 5, 11, 41$
Class number $236614400$ (GRH)
Class group $[4, 4, 20, 739420]$ (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![772639242001, -630321392910, 397271646145, -87476668070, 14466896125, -599153951, 21057190, -266090, 4510, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 4510*x^8 - 266090*x^7 + 21057190*x^6 - 599153951*x^5 + 14466896125*x^4 - 87476668070*x^3 + 397271646145*x^2 - 630321392910*x + 772639242001)
 
gp: K = bnfinit(x^10 + 4510*x^8 - 266090*x^7 + 21057190*x^6 - 599153951*x^5 + 14466896125*x^4 - 87476668070*x^3 + 397271646145*x^2 - 630321392910*x + 772639242001, 1)
 

Normalized defining polynomial

\( x^{10} + 4510 x^{8} - 266090 x^{7} + 21057190 x^{6} - 599153951 x^{5} + 14466896125 x^{4} - 87476668070 x^{3} + 397271646145 x^{2} - 630321392910 x + 772639242001 \)

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:  \(-63465642057324252347571258544921875=-\,3^{5}\cdot 5^{16}\cdot 11^{8}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3021.72$
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:  $3, 5, 11, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(33825=3\cdot 5^{2}\cdot 11\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{33825}(1,·)$, $\chi_{33825}(4321,·)$, $\chi_{33825}(4711,·)$, $\chi_{33825}(4856,·)$, $\chi_{33825}(10916,·)$, $\chi_{33825}(11276,·)$, $\chi_{33825}(15596,·)$, $\chi_{33825}(15986,·)$, $\chi_{33825}(27406,·)$, $\chi_{33825}(33466,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{4059} a^{5} - \frac{1}{9} a^{4} - \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{36531} a^{6} - \frac{2}{36531} a^{5} + \frac{11}{81} a^{4} + \frac{4}{81} a^{3} - \frac{40}{81} a^{2} + \frac{37}{81} a + \frac{34}{81}$, $\frac{1}{2447577} a^{7} - \frac{4}{2447577} a^{6} - \frac{100}{815859} a^{5} - \frac{53}{603} a^{4} + \frac{146}{1809} a^{3} + \frac{250}{603} a^{2} + \frac{770}{5427} a - \frac{932}{5427}$, $\frac{1}{374479281} a^{8} - \frac{1}{5589243} a^{7} - \frac{3599}{374479281} a^{6} - \frac{22697}{374479281} a^{5} + \frac{34844}{830331} a^{4} + \frac{40048}{830331} a^{3} + \frac{120454}{830331} a^{2} + \frac{256949}{830331} a - \frac{403919}{830331}$, $\frac{1}{527595435506311309096332565401} a^{9} + \frac{166643205795919838}{270700582609703083168975149} a^{8} - \frac{27966792504655378254853}{527595435506311309096332565401} a^{7} + \frac{1723170609267356357599298}{175865145168770436365444188467} a^{6} + \frac{38405668705592673734100883}{527595435506311309096332565401} a^{5} + \frac{188350417974043060511707364}{1169834668528406450324462451} a^{4} - \frac{27180425509185690490965070}{389944889509468816774820817} a^{3} + \frac{344781279634033233799747234}{1169834668528406450324462451} a^{2} - \frac{11396303372696614497628367}{68813804031082732372027203} a - \frac{171959491627397255477249}{600223021307545638955599}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{20}\times C_{739420}$, which has order $236614400$ (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:  \( \frac{842859542338400}{383148464419979164194867513} a^{9} + \frac{59376545330}{17871564178365556424967} a^{8} + \frac{3808193113781160700}{383148464419979164194867513} a^{7} - \frac{72832418160663815710}{127716154806659721398289171} a^{6} + \frac{17441479959103600408337}{383148464419979164194867513} a^{5} - \frac{1063981823929090776200}{849553136186206572494163} a^{4} + \frac{8551282231692146735450}{283184378728735524164721} a^{3} - \frac{130294282845721803197365}{849553136186206572494163} a^{2} + \frac{704257403252812850725630}{849553136186206572494163} a - \frac{137410773951270133513}{435891809228428205487} \) (order $6$)
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

\(\Q(\sqrt{-3}) \), 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 ${\href{/LocalNumberField/2.10.0.1}{10} }$ R R ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.10.16.9$x^{10} + 40 x^{9} + 10 x^{8} + 70 x^{7} + 15 x^{6} + 70 x^{5} + 5 x^{4} - 45 x^{3} - 20 x^{2} + 90 x + 7$$5$$2$$16$$C_{10}$$[2]^{2}$
$11$11.10.8.4$x^{10} - 781 x^{5} + 290521$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$41$41.10.8.3$x^{10} + 943 x^{5} + 242064$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$