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Properties

Label 10.10.5121551119...1493.1
Degree $10$
Signature $[10, 0]$
Discriminant $3^{5}\cdot 19^{5}\cdot 37^{5}\cdot 2617^{5}$
Root discriminant $2349.31$
Ramified primes $3, 19, 37, 2617$
Class number $48$ (GRH)
Class group $[2, 2, 2, 6]$ (GRH)
Galois group $S_5\times C_2$ (as 10T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-224574867150077989413, -210143682860212395, 163548767062839187, 40223478097149, -32841471350544, -3083201570, 2569320032, 125379, -84935, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 84935*x^8 + 125379*x^7 + 2569320032*x^6 - 3083201570*x^5 - 32841471350544*x^4 + 40223478097149*x^3 + 163548767062839187*x^2 - 210143682860212395*x - 224574867150077989413)
 
gp: K = bnfinit(x^10 - 2*x^9 - 84935*x^8 + 125379*x^7 + 2569320032*x^6 - 3083201570*x^5 - 32841471350544*x^4 + 40223478097149*x^3 + 163548767062839187*x^2 - 210143682860212395*x - 224574867150077989413, 1)
 

Normalized defining polynomial

\( x^{10} - 2 x^{9} - 84935 x^{8} + 125379 x^{7} + 2569320032 x^{6} - 3083201570 x^{5} - 32841471350544 x^{4} + 40223478097149 x^{3} + 163548767062839187 x^{2} - 210143682860212395 x - 224574867150077989413 \)

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:  \(5121551119721950730382028224671493=3^{5}\cdot 19^{5}\cdot 37^{5}\cdot 2617^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2349.31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{17107712885445664584122440752949930048909254470916288763675733148753318145521} a^{9} - \frac{4896481675452672833807419941498570510171141020745549781167215430536671631732}{17107712885445664584122440752949930048909254470916288763675733148753318145521} a^{8} - \frac{4844935933638298365026082825291385810687323157753437244558546767003422643011}{17107712885445664584122440752949930048909254470916288763675733148753318145521} a^{7} - \frac{2548026188679311547562950069171078570236391250003417816087957249215348028584}{5702570961815221528040813584316643349636418156972096254558577716251106048507} a^{6} + \frac{297402830709095455725974381666522944626257133309112906682363758645518089941}{743813603715028894961845250128257828213445846561577772333727528206666006327} a^{5} + \frac{3569096424424997854748505096835377605689224364351114764145200209522716397305}{17107712885445664584122440752949930048909254470916288763675733148753318145521} a^{4} - \frac{32895673800865030978657238914898531053626387822127490828250475267299132107}{247937867905009631653948416709419276071148615520525924111242509402222002109} a^{3} + \frac{1426832744884880144404071837041687270057134257986522277791286055820257690311}{5702570961815221528040813584316643349636418156972096254558577716251106048507} a^{2} - \frac{7389998699255075247903612027357520026776161684354792394948407235568175091141}{17107712885445664584122440752949930048909254470916288763675733148753318145521} a + \frac{1610266104229216133692798916857323809565995910489257080073038192012076566148}{5702570961815221528040813584316643349636418156972096254558577716251106048507}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}$, which has order $48$ (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:  \( 1621704091710 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_5$ (as 10T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 14 conjugacy class representatives for $S_5\times C_2$
Character table for $S_5\times C_2$

Intermediate fields

\(\Q(\sqrt{5519253}) \), 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 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 siblings: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }$ R ${\href{/LocalNumberField/5.10.0.1}{10} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$37$37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{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_37_2617.2t1.1c1$1$ $ 3 \cdot 19 \cdot 37 \cdot 2617 $ $x^{2} - x - 1379813$ $C_2$ (as 2T1) $1$ $1$
1.3_19_2617.2t1.1c1$1$ $ 3 \cdot 19 \cdot 2617 $ $x^{2} - x - 37292$ $C_2$ (as 2T1) $1$ $1$
1.37.2t1.1c1$1$ $ 37 $ $x^{2} - x - 9$ $C_2$ (as 2T1) $1$ $1$
* 4.3e3_19e3_37e4_2617e3.10t22.1c1$4$ $ 3^{3} \cdot 19^{3} \cdot 37^{4} \cdot 2617^{3}$ $x^{10} - 2 x^{9} - 84935 x^{8} + 125379 x^{7} + 2569320032 x^{6} - 3083201570 x^{5} - 32841471350544 x^{4} + 40223478097149 x^{3} + 163548767062839187 x^{2} - 210143682860212395 x - 224574867150077989413$ $S_5\times C_2$ (as 10T22) $1$ $4$
4.3_19_37e4_2617.10t22.1c1$4$ $ 3 \cdot 19 \cdot 37^{4} \cdot 2617 $ $x^{10} - 2 x^{9} - 84935 x^{8} + 125379 x^{7} + 2569320032 x^{6} - 3083201570 x^{5} - 32841471350544 x^{4} + 40223478097149 x^{3} + 163548767062839187 x^{2} - 210143682860212395 x - 224574867150077989413$ $S_5\times C_2$ (as 10T22) $1$ $4$
* 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) $1$ $4$
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) $1$ $4$
5.3e2_19e2_37e5_2617e2.12t123.1c1$5$ $ 3^{2} \cdot 19^{2} \cdot 37^{5} \cdot 2617^{2}$ $x^{10} - 2 x^{9} - 84935 x^{8} + 125379 x^{7} + 2569320032 x^{6} - 3083201570 x^{5} - 32841471350544 x^{4} + 40223478097149 x^{3} + 163548767062839187 x^{2} - 210143682860212395 x - 224574867150077989413$ $S_5\times C_2$ (as 10T22) $1$ $5$
5.3e3_19e3_37e5_2617e3.12t123.1c1$5$ $ 3^{3} \cdot 19^{3} \cdot 37^{5} \cdot 2617^{3}$ $x^{10} - 2 x^{9} - 84935 x^{8} + 125379 x^{7} + 2569320032 x^{6} - 3083201570 x^{5} - 32841471350544 x^{4} + 40223478097149 x^{3} + 163548767062839187 x^{2} - 210143682860212395 x - 224574867150077989413$ $S_5\times C_2$ (as 10T22) $1$ $5$
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) $1$ $5$
5.3e2_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) $1$ $5$
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) $1$ $6$
6.3e3_19e3_37e6_2617e3.20t65.1c1$6$ $ 3^{3} \cdot 19^{3} \cdot 37^{6} \cdot 2617^{3}$ $x^{10} - 2 x^{9} - 84935 x^{8} + 125379 x^{7} + 2569320032 x^{6} - 3083201570 x^{5} - 32841471350544 x^{4} + 40223478097149 x^{3} + 163548767062839187 x^{2} - 210143682860212395 x - 224574867150077989413$ $S_5\times C_2$ (as 10T22) $1$ $6$

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 *.