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Properties

Label 15.1.16358756351...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 3^{13}\cdot 5^{10}\cdot 29^{13}$
Root discriminant $222.63$
Ramified primes $2, 3, 5, 29$
Class number $30$ (GRH)
Class group $[30]$ (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![-74596195, 149601655, -96560678, 362562, 26986175, -10364275, -522869, 1090677, -160455, -19903, 337, 2095, -144, -28, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 - 28*x^13 - 144*x^12 + 2095*x^11 + 337*x^10 - 19903*x^9 - 160455*x^8 + 1090677*x^7 - 522869*x^6 - 10364275*x^5 + 26986175*x^4 + 362562*x^3 - 96560678*x^2 + 149601655*x - 74596195)
 
gp: K = bnfinit(x^15 - 7*x^14 - 28*x^13 - 144*x^12 + 2095*x^11 + 337*x^10 - 19903*x^9 - 160455*x^8 + 1090677*x^7 - 522869*x^6 - 10364275*x^5 + 26986175*x^4 + 362562*x^3 - 96560678*x^2 + 149601655*x - 74596195, 1)
 

Normalized defining polynomial

\( x^{15} - 7 x^{14} - 28 x^{13} - 144 x^{12} + 2095 x^{11} + 337 x^{10} - 19903 x^{9} - 160455 x^{8} + 1090677 x^{7} - 522869 x^{6} - 10364275 x^{5} + 26986175 x^{4} + 362562 x^{3} - 96560678 x^{2} + 149601655 x - 74596195 \)

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:  \(-163587563515302975177730470000000000=-\,2^{10}\cdot 3^{13}\cdot 5^{10}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $222.63$
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:  $2, 3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{6} - \frac{1}{6} a^{3} + \frac{1}{6}$, $\frac{1}{18} a^{10} + \frac{1}{18} a^{8} + \frac{1}{3} a^{7} - \frac{5}{18} a^{6} + \frac{5}{18} a^{4} - \frac{1}{18} a^{2} + \frac{5}{18}$, $\frac{1}{18} a^{11} + \frac{1}{18} a^{9} + \frac{7}{18} a^{7} - \frac{1}{3} a^{6} + \frac{5}{18} a^{5} - \frac{1}{18} a^{3} + \frac{1}{3} a^{2} - \frac{7}{18} a + \frac{1}{3}$, $\frac{1}{54} a^{12} - \frac{1}{54} a^{11} - \frac{1}{54} a^{10} - \frac{2}{27} a^{9} - \frac{2}{27} a^{8} - \frac{8}{27} a^{7} - \frac{1}{18} a^{6} - \frac{5}{54} a^{5} - \frac{11}{54} a^{4} + \frac{5}{27} a^{3} + \frac{8}{27} a^{2} - \frac{7}{27} a + \frac{13}{27}$, $\frac{1}{54} a^{13} + \frac{1}{54} a^{11} + \frac{1}{54} a^{10} + \frac{2}{27} a^{9} + \frac{2}{27} a^{8} + \frac{1}{27} a^{7} + \frac{7}{54} a^{6} - \frac{1}{54} a^{5} - \frac{25}{54} a^{4} + \frac{7}{27} a^{3} - \frac{2}{27} a^{2} - \frac{1}{2} a + \frac{11}{54}$, $\frac{1}{42299809901254664515398718962258053958} a^{14} + \frac{51555973784364171754928973248703493}{7049968316875777419233119827043008993} a^{13} + \frac{30139243629064359058055181735881365}{21149904950627332257699359481129026979} a^{12} + \frac{10569055278596337445926987945033916}{783329812986197491025902203004778777} a^{11} + \frac{185489894315848025756176470038023214}{7049968316875777419233119827043008993} a^{10} + \frac{127007761746724552139451981778824589}{1566659625972394982051804406009557554} a^{9} + \frac{3276570204746168520298778693539722853}{42299809901254664515398718962258053958} a^{8} + \frac{15478988428518346517355059279350633}{829408037279503225792131744358001058} a^{7} + \frac{6115711852272295382853487536805197772}{21149904950627332257699359481129026979} a^{6} + \frac{702973161808903260096899116937344693}{7049968316875777419233119827043008993} a^{5} + \frac{2154478900899238232657590277377258559}{7049968316875777419233119827043008993} a^{4} - \frac{6284586167954209982999094880534456783}{14099936633751554838466239654086017986} a^{3} + \frac{6657071017529562813438069430969430426}{21149904950627332257699359481129026979} a^{2} - \frac{2218712287116351634016011685607741503}{14099936633751554838466239654086017986} a - \frac{2413533478161900390815998988282437787}{21149904950627332257699359481129026979}$

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

Class group and class number

$C_{30}$, which has order $30$ (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:  \( 69747520577.13919 \) (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

3.1.1740.1, 5.1.7161220125.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ R ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.5.4.1$x^{5} - 29$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
29.10.9.2$x^{10} + 58$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$