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Properties

Label 15.3.93584704678...7025.1
Degree $15$
Signature $[3, 6]$
Discriminant $3^{20}\cdot 5^{2}\cdot 13^{6}\cdot 29^{6}\cdot 193373^{2}$
Root discriminant $291.57$
Ramified primes $3, 5, 13, 29, 193373$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T77

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-18293453208700, -20873986887750, -8699479026975, -1355688777592, 79169944845, 53704370709, 5421506331, -284869677, -97420446, -4048662, 683808, 70290, -1639, -399, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 399*x^13 - 1639*x^12 + 70290*x^11 + 683808*x^10 - 4048662*x^9 - 97420446*x^8 - 284869677*x^7 + 5421506331*x^6 + 53704370709*x^5 + 79169944845*x^4 - 1355688777592*x^3 - 8699479026975*x^2 - 20873986887750*x - 18293453208700)
 
gp: K = bnfinit(x^15 - 399*x^13 - 1639*x^12 + 70290*x^11 + 683808*x^10 - 4048662*x^9 - 97420446*x^8 - 284869677*x^7 + 5421506331*x^6 + 53704370709*x^5 + 79169944845*x^4 - 1355688777592*x^3 - 8699479026975*x^2 - 20873986887750*x - 18293453208700, 1)
 

Normalized defining polynomial

\( x^{15} - 399 x^{13} - 1639 x^{12} + 70290 x^{11} + 683808 x^{10} - 4048662 x^{9} - 97420446 x^{8} - 284869677 x^{7} + 5421506331 x^{6} + 53704370709 x^{5} + 79169944845 x^{4} - 1355688777592 x^{3} - 8699479026975 x^{2} - 20873986887750 x - 18293453208700 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9358470467826907480292426652317657025=3^{20}\cdot 5^{2}\cdot 13^{6}\cdot 29^{6}\cdot 193373^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $291.57$
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:  $3, 5, 13, 29, 193373$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{1064816744660832146873031759387009547598997245938415025325029664659797996118176948320} a^{14} - \frac{1908740048546707030452449098030518964729287960744824779929205197387227339476390087}{212963348932166429374606351877401909519799449187683005065005932931959599223635389664} a^{13} - \frac{23991704071552236526510578434130878401717816082697882069117971444028736759956421067}{532408372330416073436515879693504773799498622969207512662514832329898998059088474160} a^{12} - \frac{107802979568758483550746112526965233794876180591539164964316142631376124145319868469}{1064816744660832146873031759387009547598997245938415025325029664659797996118176948320} a^{11} - \frac{4057621079908838320674442307499408132391551819095465298570255058495185933862554211}{212963348932166429374606351877401909519799449187683005065005932931959599223635389664} a^{10} - \frac{344502246044151669402580308728393853422785906629393056724345752621802713959551658147}{1064816744660832146873031759387009547598997245938415025325029664659797996118176948320} a^{9} - \frac{482862748610272678461609202418591693701511260232096298159665375071498204186487752157}{1064816744660832146873031759387009547598997245938415025325029664659797996118176948320} a^{8} - \frac{469728428735920819209470411124784659362469518868514397405707533505724780725831832791}{1064816744660832146873031759387009547598997245938415025325029664659797996118176948320} a^{7} - \frac{33165570633011541153512128140393739708982776844692276108587044163383060048279264339}{133102093082604018359128969923376193449874655742301878165628708082474749514772118540} a^{6} - \frac{281191939024195093139568166670247259437727602963236634192902136798971388961316644589}{1064816744660832146873031759387009547598997245938415025325029664659797996118176948320} a^{5} + \frac{82299597693499923689032255403227855128016664361232217719388561668416973860315741271}{266204186165208036718257939846752386899749311484603756331257416164949499029544237080} a^{4} - \frac{39175885195627846135407156504296043502717852918670719887420819038574161744001331051}{212963348932166429374606351877401909519799449187683005065005932931959599223635389664} a^{3} - \frac{93251675603169835257598603105026680436424107276467545436274249658095742685204374707}{1064816744660832146873031759387009547598997245938415025325029664659797996118176948320} a^{2} + \frac{5315050473076975357125894547963514405681490556638711568051585185043744559185659313}{106481674466083214687303175938700954759899724593841502532502966465979799611817694832} a + \frac{26516377448572674487552345566180397895174619708320947334383160206130951525491794895}{53240837233041607343651587969350477379949862296920751266251483232989899805908847416}$

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:  $8$
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:  \( 9163315186690 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T77:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 29160
The 48 conjugacy class representatives for 1/2[3^5:2]S(5)
Character table for 1/2[3^5:2]S(5) is not computed

Intermediate fields

5.1.10933.1

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

Sibling fields

Degree 15 siblings: data not computed
Degree 30 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{3}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ $15$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
3Data not computed
$5$5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.12.0.1$x^{12} - x^{3} - 2 x + 3$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.9.6.1$x^{9} - 841 x^{3} + 73167$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
193373Data not computed