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Properties

Label 15.15.2951913264...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 1609^{2}\cdot 30241^{4}\cdot 34326797^{2}$
Root discriminant $6781.73$
Ramified primes $2, 5, 7, 1609, 30241, 34326797$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T92

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-241928000, 1693496000, -2262026800, -4929283000, 8709408000, 2017438560, -7767061528, 2232940764, 706537368, -182612916, -1229584, 1066084, -2232, -1980, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 1980*x^13 - 2232*x^12 + 1066084*x^11 - 1229584*x^10 - 182612916*x^9 + 706537368*x^8 + 2232940764*x^7 - 7767061528*x^6 + 2017438560*x^5 + 8709408000*x^4 - 4929283000*x^3 - 2262026800*x^2 + 1693496000*x - 241928000)
 
gp: K = bnfinit(x^15 - 1980*x^13 - 2232*x^12 + 1066084*x^11 - 1229584*x^10 - 182612916*x^9 + 706537368*x^8 + 2232940764*x^7 - 7767061528*x^6 + 2017438560*x^5 + 8709408000*x^4 - 4929283000*x^3 - 2262026800*x^2 + 1693496000*x - 241928000, 1)
 

Normalized defining polynomial

\( x^{15} - 1980 x^{13} - 2232 x^{12} + 1066084 x^{11} - 1229584 x^{10} - 182612916 x^{9} + 706537368 x^{8} + 2232940764 x^{7} - 7767061528 x^{6} + 2017438560 x^{5} + 8709408000 x^{4} - 4929283000 x^{3} - 2262026800 x^{2} + 1693496000 x - 241928000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2951913264025688280807555730603157129805843381784576000000=2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 1609^{2}\cdot 30241^{4}\cdot 34326797^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $6781.73$
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:  $2, 5, 7, 1609, 30241, 34326797$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{20} a^{11} - \frac{1}{10} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{10} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{40} a^{12} - \frac{1}{20} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} + \frac{1}{10} a^{6} + \frac{1}{5} a^{5} + \frac{1}{10} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{115600} a^{13} - \frac{31}{5780} a^{12} - \frac{67}{2890} a^{11} + \frac{567}{28900} a^{10} - \frac{519}{28900} a^{9} + \frac{546}{7225} a^{8} - \frac{6959}{28900} a^{7} - \frac{2289}{14450} a^{6} - \frac{6049}{28900} a^{5} + \frac{1412}{7225} a^{4} + \frac{314}{1445} a^{3} + \frac{31}{289} a^{2} + \frac{5}{578} a - \frac{108}{289}$, $\frac{1}{151549956421892949959339452028020409295190100144659594498400} a^{14} + \frac{246944647521776101521777893227468162906493295400839351}{75774978210946474979669726014010204647595050072329797249200} a^{13} + \frac{35558688605109005438086316297732887667279963962095993437}{7577497821094647497966972601401020464759505007232979724920} a^{12} + \frac{473139770061873848454052546708608503824759806042398122491}{18943744552736618744917431503502551161898762518082449312300} a^{11} + \frac{294812662467380235040368385456567234139319318934272441431}{7577497821094647497966972601401020464759505007232979724920} a^{10} + \frac{236136267657845023970628325723496022674396459076003165943}{18943744552736618744917431503502551161898762518082449312300} a^{9} + \frac{1918775788472293960027882660856822773916369141179984644739}{37887489105473237489834863007005102323797525036164898624600} a^{8} + \frac{2230155756336465629405607930084467544806707513635527693271}{9471872276368309372458715751751275580949381259041224656150} a^{7} - \frac{4376629847765569352152292676575761259068696486369284813}{89147033189348794093729089428247299585405941261564467352} a^{6} + \frac{23784520880760249071428253784348755843366175017147651193}{1894374455273661874491743150350255116189876251808244931230} a^{5} + \frac{608839249418798476635953206527520658565151829592405485859}{9471872276368309372458715751751275580949381259041224656150} a^{4} + \frac{3569328578244116428892179233277234983339918539293609736}{7959556534763285186940097270379223177268387612639684585} a^{3} - \frac{512235269819877823056302729078142892330539457338581181339}{3788748910547323748983486300700510232379752503616489862460} a^{2} - \frac{8830353741683451283428935474332312098861415046114376999}{189437445527366187449174315035025511618987625180824493123} a - \frac{41342533683478215467722850326155727527480576901805975877}{189437445527366187449174315035025511618987625180824493123}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $14$
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:  \( 362134556768000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T92:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 648000
The 55 conjugacy class representatives for [A(5)^3]3=A(5)wr3 are not computed
Character table for [A(5)^3]3=A(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 sibling: data not computed
Degree 36 sibling: 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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R R $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $15$ $15$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ $15$ $15$

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$$[\ ]^{3}$
2.12.18.44$x^{12} - 6 x^{11} + 8 x^{10} + 6 x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 8$$4$$3$$18$12T166$[2, 2, 2, 2, 2, 2]^{9}$
$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}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
1609Data not computed
30241Data not computed
34326797Data not computed