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Properties

Label 15.11.8191745500...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{12}\cdot 5^{6}\cdot 7^{12}\cdot 13^{3}\cdot 113^{4}\cdot 508087^{2}$
Root discriminant $534.02$
Ramified primes $2, 5, 7, 13, 113, 508087$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T101

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![113000, 395500, -734500, -3014275, 600030, 5465315, 3065428, -542818, -687906, -75798, 25854, 5535, -72, -153, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 153*x^13 - 72*x^12 + 5535*x^11 + 25854*x^10 - 75798*x^9 - 687906*x^8 - 542818*x^7 + 3065428*x^6 + 5465315*x^5 + 600030*x^4 - 3014275*x^3 - 734500*x^2 + 395500*x + 113000)
 
gp: K = bnfinit(x^15 - 153*x^13 - 72*x^12 + 5535*x^11 + 25854*x^10 - 75798*x^9 - 687906*x^8 - 542818*x^7 + 3065428*x^6 + 5465315*x^5 + 600030*x^4 - 3014275*x^3 - 734500*x^2 + 395500*x + 113000, 1)
 

Normalized defining polynomial

\( x^{15} - 153 x^{13} - 72 x^{12} + 5535 x^{11} + 25854 x^{10} - 75798 x^{9} - 687906 x^{8} - 542818 x^{7} + 3065428 x^{6} + 5465315 x^{5} + 600030 x^{4} - 3014275 x^{3} - 734500 x^{2} + 395500 x + 113000 \)

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:  \(81917455001598033212058392787403072000000=2^{12}\cdot 5^{6}\cdot 7^{12}\cdot 13^{3}\cdot 113^{4}\cdot 508087^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $534.02$
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:  $2, 5, 7, 13, 113, 508087$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{1950} a^{13} + \frac{1}{195} a^{12} - \frac{31}{650} a^{11} + \frac{274}{975} a^{10} - \frac{173}{390} a^{9} - \frac{136}{325} a^{8} - \frac{304}{975} a^{7} - \frac{148}{975} a^{6} - \frac{404}{975} a^{5} - \frac{56}{975} a^{4} - \frac{107}{390} a^{3} + \frac{17}{65} a^{2} - \frac{5}{26} a - \frac{2}{39}$, $\frac{1}{494262642437594352495925725908218950900} a^{14} + \frac{45888054562840971471650609278869649}{247131321218797176247962862954109475450} a^{13} + \frac{19486665312132739020756778610392151227}{494262642437594352495925725908218950900} a^{12} - \frac{7900617512018883869269176243351636923}{247131321218797176247962862954109475450} a^{11} - \frac{129861660681525257818448392068656479661}{494262642437594352495925725908218950900} a^{10} - \frac{57407683072170661074286143516163616809}{123565660609398588123981431477054737725} a^{9} + \frac{114967126599348916704656863477860855527}{247131321218797176247962862954109475450} a^{8} + \frac{3782531217621076354916461500505973351}{49426264243759435249592572590821895090} a^{7} - \frac{9904750629130770313498440335438824511}{82377107072932392082654287651369825150} a^{6} - \frac{12023027641822562614076272970927192819}{123565660609398588123981431477054737725} a^{5} + \frac{41627575466570446779833418256727670199}{494262642437594352495925725908218950900} a^{4} - \frac{9156698461053318010767796538477246411}{24713132121879717624796286295410947545} a^{3} - \frac{746172538582574877691841559878751879}{2196722855278197122204114337369862004} a^{2} + \frac{2572774466022918042616862621805503095}{9885252848751887049918514518164379018} a - \frac{70824822953099559021063781560250465}{380202032644303348073789019929399193}$

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:  \( 2090737998620000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T101:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 133 conjugacy class representatives for [S(5)^3]3=S(5)wr3 are not computed
Character table for [S(5)^3]3=S(5)wr3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ R $15$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.12.2$x^{12} - 6 x^{10} - 13 x^{8} - 28 x^{6} + 15 x^{4} - 30 x^{2} - 3$$2$$6$$12$12T105$[2, 2, 2, 2, 2]^{6}$
$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$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113Data not computed
508087Data not computed