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Properties

Label 15.7.20986118570...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{2}\cdot 13^{2}\cdot 23\cdot 229^{5}\cdot 607^{4}\cdot 1773841^{2}$
Root discriminant $2263.58$
Ramified primes $2, 5, 7, 13, 23, 229, 607, 1773841$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T102

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9945088000, 26105856000, -16999884800, -5671808000, 7849238400, -741890240, -836759048, 184136684, -217332, -2560086, 100486, 36049, -1332, -405, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 405*x^13 - 1332*x^12 + 36049*x^11 + 100486*x^10 - 2560086*x^9 - 217332*x^8 + 184136684*x^7 - 836759048*x^6 - 741890240*x^5 + 7849238400*x^4 - 5671808000*x^3 - 16999884800*x^2 + 26105856000*x - 9945088000)
 
gp: K = bnfinit(x^15 - 405*x^13 - 1332*x^12 + 36049*x^11 + 100486*x^10 - 2560086*x^9 - 217332*x^8 + 184136684*x^7 - 836759048*x^6 - 741890240*x^5 + 7849238400*x^4 - 5671808000*x^3 - 16999884800*x^2 + 26105856000*x - 9945088000, 1)
 

Normalized defining polynomial

\( x^{15} - 405 x^{13} - 1332 x^{12} + 36049 x^{11} + 100486 x^{10} - 2560086 x^{9} - 217332 x^{8} + 184136684 x^{7} - 836759048 x^{6} - 741890240 x^{5} + 7849238400 x^{4} - 5671808000 x^{3} - 16999884800 x^{2} + 26105856000 x - 9945088000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(209861185704489525754484343252978976463040512000000=2^{18}\cdot 5^{6}\cdot 7^{2}\cdot 13^{2}\cdot 23\cdot 229^{5}\cdot 607^{4}\cdot 1773841^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2263.58$
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:  $2, 5, 7, 13, 23, 229, 607, 1773841$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{10} + \frac{3}{32} a^{8} - \frac{1}{8} a^{7} - \frac{7}{32} a^{6} - \frac{5}{16} a^{5} + \frac{1}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{640} a^{11} - \frac{1}{128} a^{9} - \frac{13}{160} a^{8} + \frac{129}{640} a^{7} + \frac{3}{320} a^{6} + \frac{157}{320} a^{5} - \frac{53}{160} a^{4} - \frac{29}{160} a^{3} - \frac{1}{80} a^{2}$, $\frac{1}{2560} a^{12} - \frac{1}{512} a^{10} - \frac{13}{640} a^{9} + \frac{129}{2560} a^{8} + \frac{163}{1280} a^{7} - \frac{163}{1280} a^{6} + \frac{27}{640} a^{5} - \frac{189}{640} a^{4} - \frac{41}{320} a^{3} - \frac{1}{2} a$, $\frac{1}{51200} a^{13} - \frac{1}{10240} a^{11} + \frac{67}{12800} a^{10} + \frac{2049}{51200} a^{9} - \frac{2557}{25600} a^{8} + \frac{157}{25600} a^{7} - \frac{2133}{12800} a^{6} - \frac{829}{12800} a^{5} - \frac{281}{6400} a^{4} - \frac{13}{40} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{2310419989340024326404229356663209894065765168680550400} a^{14} + \frac{99027221601331825708948996267517303814688383747}{14440124933375152040026433479145061837911032304253440} a^{13} + \frac{61475362705270823307292561094285985306644118032559}{462083997868004865280845871332641978813153033736110080} a^{12} + \frac{283792349209842474441024903802721177606545003514907}{577604997335006081601057339165802473516441292170137600} a^{11} - \frac{17636588931560810381758641450678387917217742948201391}{2310419989340024326404229356663209894065765168680550400} a^{10} + \frac{15146925284396237230001070161036947749096239067471283}{1155209994670012163202114678331604947032882584340275200} a^{9} - \frac{93608948750762766796641634528087244002167487544306443}{1155209994670012163202114678331604947032882584340275200} a^{8} - \frac{37981590041696583431739139347723395186925021318703053}{577604997335006081601057339165802473516441292170137600} a^{7} - \frac{134257476419417349077348789392376937369954636240862949}{577604997335006081601057339165802473516441292170137600} a^{6} - \frac{134960965175400286879866313732467910822238368848155441}{288802498667503040800528669582901236758220646085068800} a^{5} + \frac{1940010699154037652398230113932365160726013562000349}{7220062466687576020013216739572530918955516152126720} a^{4} - \frac{778367015752469859683708007223624343058233477875109}{3610031233343788010006608369786265459477758076063360} a^{3} - \frac{90159668208046882428771079685640953894711369180999}{451253904167973501250826046223283182434719759507920} a^{2} + \frac{9451102485491244542519987587509638642293259563553}{45125390416797350125082604622328318243471975950792} a + \frac{535373521487274827243083160467662853054280201047}{5640673802099668765635325577791039780433996993849}$

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:  $10$
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:  \( 221942427739000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T102:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10368000
The 140 conjugacy class representatives for [S(5)^3]S(3)=S(5)wrS(3) are not computed
Character table for [S(5)^3]S(3)=S(5)wrS(3) is not computed

Intermediate fields

3.3.229.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: 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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.6$x^{4} - 20$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.10.0.1$x^{10} - x + 7$$1$$10$$0$$C_{10}$$[\ ]^{10}$
229Data not computed
607Data not computed
1773841Data not computed