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Properties

Label 15.11.6726894232...5424.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{12}\cdot 13^{3}\cdot 29^{2}\cdot 53^{3}\cdot 229^{5}\cdot 97367^{2}$
Root discriminant $285.23$
Ramified primes $2, 13, 29, 53, 229, 97367$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T82

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1344637, 12024364, 29908420, 21891109, 4161024, -2430874, -1563298, -512976, -247562, -78190, 5664, 6782, 391, -148, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 - 148*x^13 + 391*x^12 + 6782*x^11 + 5664*x^10 - 78190*x^9 - 247562*x^8 - 512976*x^7 - 1563298*x^6 - 2430874*x^5 + 4161024*x^4 + 21891109*x^3 + 29908420*x^2 + 12024364*x - 1344637)
 
gp: K = bnfinit(x^15 - 6*x^14 - 148*x^13 + 391*x^12 + 6782*x^11 + 5664*x^10 - 78190*x^9 - 247562*x^8 - 512976*x^7 - 1563298*x^6 - 2430874*x^5 + 4161024*x^4 + 21891109*x^3 + 29908420*x^2 + 12024364*x - 1344637, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} - 148 x^{13} + 391 x^{12} + 6782 x^{11} + 5664 x^{10} - 78190 x^{9} - 247562 x^{8} - 512976 x^{7} - 1563298 x^{6} - 2430874 x^{5} + 4161024 x^{4} + 21891109 x^{3} + 29908420 x^{2} + 12024364 x - 1344637 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6726894232542085043013968658644455424=2^{12}\cdot 13^{3}\cdot 29^{2}\cdot 53^{3}\cdot 229^{5}\cdot 97367^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $285.23$
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:  $2, 13, 29, 53, 229, 97367$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{12} - \frac{1}{4}$, $\frac{1}{84} a^{13} - \frac{1}{12} a^{12} - \frac{1}{21} a^{11} - \frac{3}{14} a^{10} - \frac{1}{14} a^{9} + \frac{1}{7} a^{8} + \frac{5}{21} a^{7} + \frac{1}{6} a^{6} + \frac{2}{21} a^{5} - \frac{19}{42} a^{4} + \frac{1}{42} a^{3} - \frac{25}{84} a - \frac{1}{12}$, $\frac{1}{317700687170135294105268026455175028208367916} a^{14} + \frac{48010490068049386272139437508073838486437}{158850343585067647052634013227587514104183958} a^{13} + \frac{4406983922930377331174453151731386104205371}{35300076352237254900585336272797225356485324} a^{12} - \frac{16268574929625712779926399730081780848769545}{79425171792533823526317006613793757052091979} a^{11} + \frac{3990544391048742879693286006077850561143651}{17650038176118627450292668136398612678242662} a^{10} - \frac{3808423878690526523954843988103938639179120}{26475057264177941175439002204597919017363993} a^{9} - \frac{35436246730331133687929707934915302429247699}{158850343585067647052634013227587514104183958} a^{8} + \frac{10253712019416338097260903943575164179182581}{52950114528355882350878004409195838034727986} a^{7} + \frac{9768677514231932921299418969456402256239951}{52950114528355882350878004409195838034727986} a^{6} + \frac{32682235407422551415815425823448741240242487}{79425171792533823526317006613793757052091979} a^{5} - \frac{792931055782313112914509593864961967567183}{2997176294057880133068566287312971964229886} a^{4} - \frac{8355493811162262344511809901624442268772752}{79425171792533823526317006613793757052091979} a^{3} + \frac{21516097474745889021300387997312538445926333}{317700687170135294105268026455175028208367916} a^{2} + \frac{989540945687070750034725145653794427152477}{26475057264177941175439002204597919017363993} a + \frac{18896061575894183058977328017037091204074757}{45385812452876470586466860922167861172623988}$

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:  $12$
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:  \( 58551393935000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T82:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48000
The 65 conjugacy class representatives for [F(5)^3]S(3)=F(5)wrS(3) are not computed
Character table for [F(5)^3]S(3)=F(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 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ R ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{5}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$53$$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.4.3.4$x^{4} + 424$$4$$1$$3$$C_4$$[\ ]_{4}$
229Data not computed
97367Data not computed