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Properties

Label 15.1.11274121380...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{12}\cdot 3^{13}\cdot 5^{28}\cdot 541^{5}$
Root discriminant $741.55$
Ramified primes $2, 3, 5, 541$
Class number $15$ (GRH)
Class group $[15]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5802471126, 5757007770, -1472601870, -768468060, 446434740, 19156149, -17116785, 7842555, -418770, 55665, -9901, 2165, -460, 55, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 55*x^13 - 460*x^12 + 2165*x^11 - 9901*x^10 + 55665*x^9 - 418770*x^8 + 7842555*x^7 - 17116785*x^6 + 19156149*x^5 + 446434740*x^4 - 768468060*x^3 - 1472601870*x^2 + 5757007770*x + 5802471126)
 
gp: K = bnfinit(x^15 - 5*x^14 + 55*x^13 - 460*x^12 + 2165*x^11 - 9901*x^10 + 55665*x^9 - 418770*x^8 + 7842555*x^7 - 17116785*x^6 + 19156149*x^5 + 446434740*x^4 - 768468060*x^3 - 1472601870*x^2 + 5757007770*x + 5802471126, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 55 x^{13} - 460 x^{12} + 2165 x^{11} - 9901 x^{10} + 55665 x^{9} - 418770 x^{8} + 7842555 x^{7} - 17116785 x^{6} + 19156149 x^{5} + 446434740 x^{4} - 768468060 x^{3} - 1472601870 x^{2} + 5757007770 x + 5802471126 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-11274121380851814944915771484375000000000000=-\,2^{12}\cdot 3^{13}\cdot 5^{28}\cdot 541^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $741.55$
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:  $2, 3, 5, 541$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{15} a^{6} + \frac{2}{15} a^{4} - \frac{1}{3} a^{3} - \frac{7}{15} a^{2} - \frac{2}{5}$, $\frac{1}{15} a^{7} + \frac{2}{15} a^{5} - \frac{2}{15} a^{3} + \frac{1}{3} a^{2} - \frac{2}{5} a$, $\frac{1}{45} a^{8} - \frac{1}{45} a^{7} - \frac{7}{45} a^{5} + \frac{4}{45} a^{4} + \frac{4}{15} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{45} a^{9} - \frac{1}{45} a^{7} - \frac{1}{45} a^{6} - \frac{1}{15} a^{5} - \frac{2}{45} a^{4} + \frac{1}{3} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{675} a^{10} - \frac{1}{135} a^{9} - \frac{1}{135} a^{8} + \frac{1}{135} a^{7} + \frac{1}{135} a^{6} - \frac{17}{135} a^{5} + \frac{2}{45} a^{4} - \frac{1}{3} a^{3} + \frac{2}{15} a^{2} + \frac{1}{5} a + \frac{2}{25}$, $\frac{1}{2025} a^{11} + \frac{1}{2025} a^{10} - \frac{4}{405} a^{9} + \frac{1}{405} a^{8} - \frac{2}{405} a^{7} - \frac{1}{81} a^{6} - \frac{19}{135} a^{5} - \frac{7}{45} a^{4} + \frac{1}{3} a^{3} + \frac{1}{5} a^{2} + \frac{7}{75} a + \frac{9}{25}$, $\frac{1}{30375} a^{12} - \frac{2}{30375} a^{11} - \frac{14}{30375} a^{10} + \frac{22}{6075} a^{9} - \frac{59}{6075} a^{8} - \frac{179}{6075} a^{7} + \frac{2}{405} a^{6} - \frac{28}{225} a^{5} - \frac{13}{225} a^{4} + \frac{44}{225} a^{3} - \frac{98}{1125} a^{2} - \frac{83}{375} a + \frac{13}{125}$, $\frac{1}{30375} a^{13} - \frac{1}{10125} a^{11} + \frac{7}{30375} a^{10} + \frac{1}{405} a^{9} - \frac{19}{2025} a^{8} - \frac{178}{6075} a^{7} - \frac{17}{2025} a^{6} + \frac{26}{225} a^{5} - \frac{32}{225} a^{4} + \frac{164}{375} a^{3} - \frac{74}{225} a^{2} - \frac{92}{375} a + \frac{26}{125}$, $\frac{1}{92894342432935933497102720166814382538824375} a^{14} + \frac{747735047047950945154039979225779099666}{92894342432935933497102720166814382538824375} a^{13} - \frac{1362581701504359492013405783236562607639}{92894342432935933497102720166814382538824375} a^{12} + \frac{18948568644058134553467708944111449447961}{92894342432935933497102720166814382538824375} a^{11} - \frac{9989091276524558635503908626133604343294}{92894342432935933497102720166814382538824375} a^{10} - \frac{135507635186922206761691404192215478724804}{18578868486587186699420544033362876507764875} a^{9} + \frac{6578504131127480082665362678053870626302}{688106240243969877756316445680106537324625} a^{8} + \frac{1633959797966613267744535533227679308243}{137621248048793975551263289136021307464925} a^{7} + \frac{4926340559511155500860077598337828573123}{688106240243969877756316445680106537324625} a^{6} + \frac{12812512916302030179786358209424198832969}{229368746747989959252105481893368845774875} a^{5} - \frac{570927650256450671533060859204403128557028}{3440531201219849388781582228400532686623125} a^{4} + \frac{136713106550086726878820771791455547105698}{382281244579983265420175803155614742958125} a^{3} - \frac{15266496992284192751327068567709383457759}{127427081526661088473391934385204914319375} a^{2} - \frac{33739709809981599757940354089308867921829}{127427081526661088473391934385204914319375} a + \frac{21037412834755211999681897153972747721606}{127427081526661088473391934385204914319375}$

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

Class group and class number

$C_{15}$, which has order $15$ (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:  $7$
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:  \( 1314224105020017.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.8115.1, 5.1.2531250000.5

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

Sibling fields

Degree 30 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.5.9.2$x^{5} + 55$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.17$x^{10} + 10$$10$$1$$19$$F_{5}\times C_2$$[9/4]_{4}^{2}$
541Data not computed