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Properties

Label 15.7.19719109831...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{21}\cdot 5^{6}\cdot 13^{2}\cdot 17^{4}\cdot 37^{5}\cdot 247955177^{2}$
Root discriminant $660.18$
Ramified primes $2, 5, 13, 17, 37, 247955177$
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![-36992000, 129472000, -39766400, -221720800, 135668160, 82469440, -71519792, 7145208, 1573056, 458730, -60688, -1742, -2520, 144, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 144*x^13 - 2520*x^12 - 1742*x^11 - 60688*x^10 + 458730*x^9 + 1573056*x^8 + 7145208*x^7 - 71519792*x^6 + 82469440*x^5 + 135668160*x^4 - 221720800*x^3 - 39766400*x^2 + 129472000*x - 36992000)
 
gp: K = bnfinit(x^15 + 144*x^13 - 2520*x^12 - 1742*x^11 - 60688*x^10 + 458730*x^9 + 1573056*x^8 + 7145208*x^7 - 71519792*x^6 + 82469440*x^5 + 135668160*x^4 - 221720800*x^3 - 39766400*x^2 + 129472000*x - 36992000, 1)
 

Normalized defining polynomial

\( x^{15} + 144 x^{13} - 2520 x^{12} - 1742 x^{11} - 60688 x^{10} + 458730 x^{9} + 1573056 x^{8} + 7145208 x^{7} - 71519792 x^{6} + 82469440 x^{5} + 135668160 x^{4} - 221720800 x^{3} - 39766400 x^{2} + 129472000 x - 36992000 \)

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:  \(1971910983137006508480742693889540096000000=2^{21}\cdot 5^{6}\cdot 13^{2}\cdot 17^{4}\cdot 37^{5}\cdot 247955177^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $660.18$
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:  $2, 5, 13, 17, 37, 247955177$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{9} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{16} a^{10} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{160} a^{11} + \frac{1}{40} a^{9} - \frac{1}{8} a^{8} - \frac{11}{80} a^{7} + \frac{1}{5} a^{6} - \frac{3}{16} a^{5} - \frac{3}{20} a^{4} + \frac{3}{10} a^{3} - \frac{9}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{320} a^{12} + \frac{1}{80} a^{10} - \frac{1}{16} a^{9} - \frac{11}{160} a^{8} - \frac{3}{20} a^{7} - \frac{3}{32} a^{6} + \frac{17}{40} a^{5} + \frac{3}{20} a^{4} - \frac{9}{40} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{27200} a^{13} - \frac{13}{13600} a^{11} - \frac{41}{1360} a^{10} + \frac{489}{13600} a^{9} - \frac{361}{3400} a^{8} - \frac{673}{2720} a^{7} - \frac{823}{3400} a^{6} + \frac{877}{6800} a^{5} + \frac{1261}{3400} a^{4} + \frac{141}{340} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1030493453561681701618694456213162263723639302400} a^{14} + \frac{3886904735734300908491179887570888961653389}{257623363390420425404673614053290565930909825600} a^{13} + \frac{33204640049128223389624102945970644010291669}{64405840847605106351168403513322641482727456400} a^{12} - \frac{180098579650399980790437879999907315535619467}{128811681695210212702336807026645282965454912800} a^{11} - \frac{6321969540317254922589776757101747380368298311}{515246726780840850809347228106581131861819651200} a^{10} + \frac{155158562591306524486596598982133875902164791}{25762336339042042540467361405329056593090982560} a^{9} - \frac{1474782546405693045499013785550460472107316827}{30308630987108285341726307535681243050695273600} a^{8} - \frac{8082969509949931053681129319624945549045579223}{128811681695210212702336807026645282965454912800} a^{7} - \frac{9998271873838307559235733683538466966489373177}{128811681695210212702336807026645282965454912800} a^{6} + \frac{24409752094484138083726245120051119687265838251}{64405840847605106351168403513322641482727456400} a^{5} - \frac{6363826698014377178144938081001833932095533883}{16101460211901276587792100878330660370681864100} a^{4} + \frac{1365373344692168544259087217288682042457788287}{3220292042380255317558420175666132074136372820} a^{3} + \frac{112977464682360184748713902202248241550938281}{378857887338853566771578844196015538133690920} a^{2} + \frac{4542263367811652553324698925914819953530946}{9471447183471339169289471104900388453342273} a - \frac{4174547052659923431682539814301833873420205}{9471447183471339169289471104900388453342273}$

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:  \( 48284056165600000 \) (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

3.3.148.1

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 ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R $15$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$[\ ]_{3}^{2}$
2.6.11.11$x^{6} + 2 x^{4} + 2 x^{2} + 2$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
2.6.8.2$x^{6} + 2 x^{3} + 2 x^{2} + 6$$6$$1$$8$$S_4\times C_2$$[4/3, 4/3, 2]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
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}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.5.4.1$x^{5} - 17$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
37Data not computed
247955177Data not computed