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Properties

Label 10.2.12961461825709056.1
Degree $10$
Signature $[2, 4]$
Discriminant $2^{10}\cdot 3^{10}\cdot 11^{8}$
Root discriminant $40.86$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $M_{10}$ (as 10T31)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-249, -546, -555, 78, 102, -144, 6, 6, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 + 3*x^8 + 6*x^7 + 6*x^6 - 144*x^5 + 102*x^4 + 78*x^3 - 555*x^2 - 546*x - 249)
 
gp: K = bnfinit(x^10 - 2*x^9 + 3*x^8 + 6*x^7 + 6*x^6 - 144*x^5 + 102*x^4 + 78*x^3 - 555*x^2 - 546*x - 249, 1)
 

Normalized defining polynomial

\( x^{10} - 2 x^{9} + 3 x^{8} + 6 x^{7} + 6 x^{6} - 144 x^{5} + 102 x^{4} + 78 x^{3} - 555 x^{2} - 546 x - 249 \)

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:  \(12961461825709056=2^{10}\cdot 3^{10}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.86$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11} a^{5} - \frac{3}{11} a^{4} - \frac{1}{11} a^{3} - \frac{2}{11} a^{2} + \frac{3}{11} a + \frac{4}{11}$, $\frac{1}{22} a^{6} + \frac{1}{22} a^{4} + \frac{3}{11} a^{3} - \frac{3}{22} a^{2} + \frac{1}{11} a + \frac{1}{22}$, $\frac{1}{22} a^{7} - \frac{1}{22} a^{5} - \frac{5}{11} a^{4} - \frac{1}{22} a^{3} + \frac{3}{11} a^{2} - \frac{5}{22} a - \frac{4}{11}$, $\frac{1}{3146} a^{8} - \frac{53}{3146} a^{7} + \frac{29}{3146} a^{6} + \frac{125}{3146} a^{5} + \frac{1369}{3146} a^{4} - \frac{1471}{3146} a^{3} + \frac{91}{242} a^{2} + \frac{1311}{3146} a - \frac{181}{1573}$, $\frac{1}{15730} a^{9} - \frac{1}{15730} a^{8} - \frac{153}{15730} a^{7} + \frac{203}{15730} a^{6} - \frac{711}{15730} a^{5} - \frac{471}{3146} a^{4} - \frac{381}{1210} a^{3} + \frac{1297}{3146} a^{2} - \frac{369}{1573} a - \frac{196}{605}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 147944.210713 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$M_{10}$ (as 10T31):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 720
The 8 conjugacy class representatives for $M_{10}$
Character table for $M_{10}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: data not computed
Degree 40 sibling: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.4.4.5$x^{4} + 2 x + 2$$4$$1$$4$$S_4$$[4/3, 4/3]_{3}^{2}$
2.6.6.8$x^{6} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.9.10.1$x^{9} + 3 x^{2} + 3$$9$$1$$10$$C_3^2:Q_8$$[5/4, 5/4]_{4}^{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.8.7.2$x^{8} - 11$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$