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Properties

Label 10.2.82228769593...0000.1
Degree $10$
Signature $[2, 4]$
Discriminant $2^{19}\cdot 3^{8}\cdot 5^{7}\cdot 7^{8}\cdot 11^{8}\cdot 19^{5}$
Root discriminant $3903.93$
Ramified primes $2, 3, 5, 7, 11, 19$
Class number $1288000$ (GRH)
Class group $[10, 10, 12880]$ (GRH)
Galois group $F_{5}\times C_2$ (as 10T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-147164007376, -114412452000, 6516050000, -1204341600, -68590000, -633864, 361000, 0, -950, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 950*x^8 + 361000*x^6 - 633864*x^5 - 68590000*x^4 - 1204341600*x^3 + 6516050000*x^2 - 114412452000*x - 147164007376)
 
gp: K = bnfinit(x^10 - 950*x^8 + 361000*x^6 - 633864*x^5 - 68590000*x^4 - 1204341600*x^3 + 6516050000*x^2 - 114412452000*x - 147164007376, 1)
 

Normalized defining polynomial

\( x^{10} - 950 x^{8} + 361000 x^{6} - 633864 x^{5} - 68590000 x^{4} - 1204341600 x^{3} + 6516050000 x^{2} - 114412452000 x - 147164007376 \)

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:  \(822287695938095069212060016640000000=2^{19}\cdot 3^{8}\cdot 5^{7}\cdot 7^{8}\cdot 11^{8}\cdot 19^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3903.93$
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:  $2, 3, 5, 7, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{70} a^{4} + \frac{1}{10} a^{3} - \frac{8}{35} a^{2} + \frac{2}{5} a - \frac{17}{35}$, $\frac{1}{140} a^{5} - \frac{3}{14} a^{3} - \frac{1}{7} a + \frac{1}{5}$, $\frac{1}{980} a^{6} + \frac{1}{245} a^{4} + \frac{1}{10} a^{3} + \frac{34}{245} a^{2} + \frac{26}{245}$, $\frac{1}{4900} a^{7} + \frac{1}{4900} a^{6} + \frac{1}{1225} a^{5} - \frac{1}{490} a^{4} + \frac{43}{490} a^{3} + \frac{482}{1225} a^{2} + \frac{222}{1225} a - \frac{247}{1225}$, $\frac{1}{68600} a^{8} - \frac{4}{8575} a^{6} + \frac{1}{350} a^{5} - \frac{2}{343} a^{4} + \frac{33}{350} a^{3} - \frac{71}{343} a^{2} - \frac{3}{175} a - \frac{3359}{8575}$, $\frac{1}{6880418188167727353231940600} a^{9} - \frac{14806458430259643949327}{3440209094083863676615970300} a^{8} + \frac{1949310391478077516929}{68804181881677273532319406} a^{7} - \frac{148379989032544779425383}{860052273520965919153992575} a^{6} - \frac{2518588172696468000937082}{860052273520965919153992575} a^{5} + \frac{6718865401241992650100677}{1720104547041931838307985150} a^{4} + \frac{285822267788379093062881827}{1720104547041931838307985150} a^{3} + \frac{78472476734940506709713274}{172010454704193183830798515} a^{2} + \frac{219274415297819586267934771}{860052273520965919153992575} a - \frac{395777036258310596381702121}{860052273520965919153992575}$

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

Class group and class number

$C_{10}\times C_{10}\times C_{12880}$, which has order $1288000$ (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:  \( 370200813.58666074 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 10T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $F_{5}\times C_2$
Character table for $F_{5}\times C_2$

Intermediate fields

\(\Q(\sqrt{190}) \), 5.1.5694792642000.6

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 10 sibling: data not computed
Degree 20 siblings: 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 R R R R R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$2$2.10.19.33$x^{10} - 6 x^{4} + 4 x^{2} - 14$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$