Properties

Label 10.0.49064203594375.1
Degree $10$
Signature $[0, 5]$
Discriminant $-4.906\times 10^{13}$
Root discriminant $23.39$
Ramified primes $5, 151$
Class number $7$
Class group $[7]$
Galois group $A_5\times C_2$ (as 10T11)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 + 7*x^8 - 18*x^7 + 79*x^6 - 205*x^5 + 297*x^4 - 250*x^3 + 130*x^2 - 45*x + 9)
 
gp: K = bnfinit(x^10 - 4*x^9 + 7*x^8 - 18*x^7 + 79*x^6 - 205*x^5 + 297*x^4 - 250*x^3 + 130*x^2 - 45*x + 9, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -45, 130, -250, 297, -205, 79, -18, 7, -4, 1]);
 

\(x^{10} - 4 x^{9} + 7 x^{8} - 18 x^{7} + 79 x^{6} - 205 x^{5} + 297 x^{4} - 250 x^{3} + 130 x^{2} - 45 x + 9\)  Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $10$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-49064203594375\)\(\medspace = -\,5^{4}\cdot 151^{5}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $23.39$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 151$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{17} a^{8} - \frac{8}{17} a^{7} - \frac{8}{17} a^{6} - \frac{1}{17} a^{5} - \frac{5}{17} a^{3} - \frac{6}{17} a^{2} - \frac{8}{17} a + \frac{2}{17}$, $\frac{1}{51} a^{9} - \frac{1}{51} a^{8} + \frac{4}{51} a^{7} - \frac{2}{17} a^{6} + \frac{10}{51} a^{5} - \frac{22}{51} a^{4} - \frac{8}{17} a^{3} - \frac{16}{51} a^{2} - \frac{20}{51} a - \frac{1}{17}$  Toggle raw display

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

Class group and class number

$C_{7}$, which has order $7$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $4$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 130.469796492 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{5}\cdot 130.469796492 \cdot 7}{2\sqrt{49064203594375}}\approx 0.638403110262$

Galois group

$C_2\times A_5$ (as 10T11):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 120
The 10 conjugacy class representatives for $A_5\times C_2$
Character table for $A_5\times C_2$

Intermediate fields

\(\Q(\sqrt{-151}) \), 5.1.570025.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/padicField/2.5.0.1}{5} }^{2}$ ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ R ${\href{/padicField/7.10.0.1}{10} }$ ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }$ ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }$ ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$151$151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.4.2.1$x^{4} + 3473 x^{2} + 3283344$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
151.4.2.1$x^{4} + 3473 x^{2} + 3283344$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.151.2t1.a.a$1$ $ 151 $ \(\Q(\sqrt{-151}) \) $C_2$ (as 2T1) $1$ $-1$
3.570025.12t33.a.a$3$ $ 5^{2} \cdot 151^{2}$ 5.1.570025.1 $A_5$ (as 5T4) $1$ $-1$
3.570025.12t33.a.b$3$ $ 5^{2} \cdot 151^{2}$ 5.1.570025.1 $A_5$ (as 5T4) $1$ $-1$
3.3775.12t76.a.a$3$ $ 5^{2} \cdot 151 $ 10.0.49064203594375.1 $A_5\times C_2$ (as 10T11) $1$ $1$
3.3775.12t76.a.b$3$ $ 5^{2} \cdot 151 $ 10.0.49064203594375.1 $A_5\times C_2$ (as 10T11) $1$ $1$
* 4.570025.5t4.a.a$4$ $ 5^{2} \cdot 151^{2}$ 5.1.570025.1 $A_5$ (as 5T4) $1$ $0$
* 4.570025.10t11.a.a$4$ $ 5^{2} \cdot 151^{2}$ 10.0.49064203594375.1 $A_5\times C_2$ (as 10T11) $1$ $0$
5.14250625.6t12.a.a$5$ $ 5^{4} \cdot 151^{2}$ 5.1.570025.1 $A_5$ (as 5T4) $1$ $1$
5.2151844375.12t75.a.a$5$ $ 5^{4} \cdot 151^{3}$ 10.0.49064203594375.1 $A_5\times C_2$ (as 10T11) $1$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.