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Properties

Label 10.10.1384203005...8689.1
Degree $10$
Signature $[10, 0]$
Discriminant $3^{5}\cdot 19^{5}\cdot 37^{4}\cdot 2617^{5}$
Root discriminant $1637.27$
Ramified primes $3, 19, 37, 2617$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $(C_2^4:A_5) : C_2$ (as 10T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-54632779228646021223, -48333574490314266, 45393680147997906, 18509469846030, -12038380228757, -2614119839, 1306127781, 170352, -60538, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 - 60538*x^8 + 170352*x^7 + 1306127781*x^6 - 2614119839*x^5 - 12038380228757*x^4 + 18509469846030*x^3 + 45393680147997906*x^2 - 48333574490314266*x - 54632779228646021223)
 
gp: K = bnfinit(x^10 - 4*x^9 - 60538*x^8 + 170352*x^7 + 1306127781*x^6 - 2614119839*x^5 - 12038380228757*x^4 + 18509469846030*x^3 + 45393680147997906*x^2 - 48333574490314266*x - 54632779228646021223, 1)
 

Normalized defining polynomial

\( x^{10} - 4 x^{9} - 60538 x^{8} + 170352 x^{7} + 1306127781 x^{6} - 2614119839 x^{5} - 12038380228757 x^{4} + 18509469846030 x^{3} + 45393680147997906 x^{2} - 48333574490314266 x - 54632779228646021223 \)

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

Invariants

Degree:  $10$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(138420300533025695415730492558689=3^{5}\cdot 19^{5}\cdot 37^{4}\cdot 2617^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1637.27$
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:  $3, 19, 37, 2617$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{56119763409756369497644344272276768675107030311699737062616576796225808701} a^{9} + \frac{2124409183489058496308175388385760732379779861275498285237715130315295413}{18706587803252123165881448090758922891702343437233245687538858932075269567} a^{8} - \frac{12783155804969803775741517922055786000558165928571020299167970340809285393}{56119763409756369497644344272276768675107030311699737062616576796225808701} a^{7} - \frac{15683943166148953390225425939343394548154447582569611268956107507695677760}{56119763409756369497644344272276768675107030311699737062616576796225808701} a^{6} + \frac{8320941246530122905026912811061883889047244103829013724379572556693302369}{18706587803252123165881448090758922891702343437233245687538858932075269567} a^{5} - \frac{21450778768840217118898916422399735885803875050340727213991391569967814245}{56119763409756369497644344272276768675107030311699737062616576796225808701} a^{4} - \frac{19407317316325904409350798252226468509605188632110457835400585898905149664}{56119763409756369497644344272276768675107030311699737062616576796225808701} a^{3} - \frac{1789390100890920593482282961972265273014464149398063687895320645615671271}{56119763409756369497644344272276768675107030311699737062616576796225808701} a^{2} + \frac{2857398920105659844581355203328029657239328140615452795480731407623021250}{18706587803252123165881448090758922891702343437233245687538858932075269567} a + \frac{2237676610216304149128841419200510081487805764862138676052409084277069987}{18706587803252123165881448090758922891702343437233245687538858932075269567}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  $9$
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:  \( 787240108562 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^4:A_5) : C_2$ (as 10T38):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 18 conjugacy class representatives for $(C_2^4:A_5) : C_2$
Character table for $(C_2^4:A_5) : C_2$

Intermediate fields

5.5.149169.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 sibling: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.2.1.1$x^{2} - 19$$2$$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$$[\ ]_{2}^{2}$
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
2617Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3_19_2617.2t1.1c1$1$ $ 3 \cdot 19 \cdot 2617 $ $x^{2} - x - 37292$ $C_2$ (as 2T1) $1$ $1$
4.3e3_19e3_2617e3.10t12.1c1$4$ $ 3^{3} \cdot 19^{3} \cdot 2617^{3}$ $x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1$ $S_5$ (as 5T5) $1$ $4$
* 4.3_19_2617.5t5.1c1$4$ $ 3 \cdot 19 \cdot 2617 $ $x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1$ $S_5$ (as 5T5) $1$ $4$
5.3e2_19e2_2617e2.10t13.1c1$5$ $ 3^{2} \cdot 19^{2} \cdot 2617^{2}$ $x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1$ $S_5$ (as 5T5) $1$ $5$
5.3_19_37e4_2617.10t37.1c1$5$ $ 3 \cdot 19 \cdot 37^{4} \cdot 2617 $ $x^{10} - 4 x^{9} - 60538 x^{8} + 170352 x^{7} + 1306127781 x^{6} - 2614119839 x^{5} - 12038380228757 x^{4} + 18509469846030 x^{3} + 45393680147997906 x^{2} - 48333574490314266 x - 54632779228646021223$ $(C_2^4:A_5) : C_2$ (as 10T38) $1$ $5$
5.3e3_19e3_2617e3.6t14.1c1$5$ $ 3^{3} \cdot 19^{3} \cdot 2617^{3}$ $x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1$ $S_5$ (as 5T5) $1$ $5$
* 5.3e4_19e4_37e4_2617e4.10t38.1c1$5$ $ 3^{4} \cdot 19^{4} \cdot 37^{4} \cdot 2617^{4}$ $x^{10} - 4 x^{9} - 60538 x^{8} + 170352 x^{7} + 1306127781 x^{6} - 2614119839 x^{5} - 12038380228757 x^{4} + 18509469846030 x^{3} + 45393680147997906 x^{2} - 48333574490314266 x - 54632779228646021223$ $(C_2^4:A_5) : C_2$ (as 10T38) $1$ $5$
6.3e3_19e3_2617e3.20t35.1c1$6$ $ 3^{3} \cdot 19^{3} \cdot 2617^{3}$ $x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1$ $S_5$ (as 5T5) $1$ $6$
10.3e6_19e6_37e4_2617e6.20t218.1c1$10$ $ 3^{6} \cdot 19^{6} \cdot 37^{4} \cdot 2617^{6}$ $x^{10} - 4 x^{9} - 60538 x^{8} + 170352 x^{7} + 1306127781 x^{6} - 2614119839 x^{5} - 12038380228757 x^{4} + 18509469846030 x^{3} + 45393680147997906 x^{2} - 48333574490314266 x - 54632779228646021223$ $(C_2^4:A_5) : C_2$ (as 10T38) $1$ $10$
10.3e4_19e4_37e4_2617e4.20t223.1c1$10$ $ 3^{4} \cdot 19^{4} \cdot 37^{4} \cdot 2617^{4}$ $x^{10} - 4 x^{9} - 60538 x^{8} + 170352 x^{7} + 1306127781 x^{6} - 2614119839 x^{5} - 12038380228757 x^{4} + 18509469846030 x^{3} + 45393680147997906 x^{2} - 48333574490314266 x - 54632779228646021223$ $(C_2^4:A_5) : C_2$ (as 10T38) $1$ $10$
10.3e7_19e7_37e4_2617e7.20t222.1c1$10$ $ 3^{7} \cdot 19^{7} \cdot 37^{4} \cdot 2617^{7}$ $x^{10} - 4 x^{9} - 60538 x^{8} + 170352 x^{7} + 1306127781 x^{6} - 2614119839 x^{5} - 12038380228757 x^{4} + 18509469846030 x^{3} + 45393680147997906 x^{2} - 48333574490314266 x - 54632779228646021223$ $(C_2^4:A_5) : C_2$ (as 10T38) $1$ $10$
10.3e5_19e5_37e8_2617e5.30t329.1c1$10$ $ 3^{5} \cdot 19^{5} \cdot 37^{8} \cdot 2617^{5}$ $x^{10} - 4 x^{9} - 60538 x^{8} + 170352 x^{7} + 1306127781 x^{6} - 2614119839 x^{5} - 12038380228757 x^{4} + 18509469846030 x^{3} + 45393680147997906 x^{2} - 48333574490314266 x - 54632779228646021223$ $(C_2^4:A_5) : C_2$ (as 10T38) $1$ $10$
10.3e3_19e3_37e4_2617e3.16t1328.1c1$10$ $ 3^{3} \cdot 19^{3} \cdot 37^{4} \cdot 2617^{3}$ $x^{10} - 4 x^{9} - 60538 x^{8} + 170352 x^{7} + 1306127781 x^{6} - 2614119839 x^{5} - 12038380228757 x^{4} + 18509469846030 x^{3} + 45393680147997906 x^{2} - 48333574490314266 x - 54632779228646021223$ $(C_2^4:A_5) : C_2$ (as 10T38) $1$ $10$
15.3e6_19e6_37e12_2617e6.30t333.1c1$15$ $ 3^{6} \cdot 19^{6} \cdot 37^{12} \cdot 2617^{6}$ $x^{10} - 4 x^{9} - 60538 x^{8} + 170352 x^{7} + 1306127781 x^{6} - 2614119839 x^{5} - 12038380228757 x^{4} + 18509469846030 x^{3} + 45393680147997906 x^{2} - 48333574490314266 x - 54632779228646021223$ $(C_2^4:A_5) : C_2$ (as 10T38) $1$ $15$
15.3e9_19e9_37e12_2617e9.30t332.1c1$15$ $ 3^{9} \cdot 19^{9} \cdot 37^{12} \cdot 2617^{9}$ $x^{10} - 4 x^{9} - 60538 x^{8} + 170352 x^{7} + 1306127781 x^{6} - 2614119839 x^{5} - 12038380228757 x^{4} + 18509469846030 x^{3} + 45393680147997906 x^{2} - 48333574490314266 x - 54632779228646021223$ $(C_2^4:A_5) : C_2$ (as 10T38) $1$ $15$
20.3e9_19e9_37e8_2617e9.40t1677.1c1$20$ $ 3^{9} \cdot 19^{9} \cdot 37^{8} \cdot 2617^{9}$ $x^{10} - 4 x^{9} - 60538 x^{8} + 170352 x^{7} + 1306127781 x^{6} - 2614119839 x^{5} - 12038380228757 x^{4} + 18509469846030 x^{3} + 45393680147997906 x^{2} - 48333574490314266 x - 54632779228646021223$ $(C_2^4:A_5) : C_2$ (as 10T38) $1$ $20$
20.3e11_19e11_37e8_2617e11.60.1c1$20$ $ 3^{11} \cdot 19^{11} \cdot 37^{8} \cdot 2617^{11}$ $x^{10} - 4 x^{9} - 60538 x^{8} + 170352 x^{7} + 1306127781 x^{6} - 2614119839 x^{5} - 12038380228757 x^{4} + 18509469846030 x^{3} + 45393680147997906 x^{2} - 48333574490314266 x - 54632779228646021223$ $(C_2^4:A_5) : C_2$ (as 10T38) $1$ $20$

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 *.