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Properties

Label 15.15.8207583591...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 5^{18}\cdot 19^{12}\cdot 37^{7}$
Root discriminant $622.70$
Ramified primes $2, 5, 19, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $((C_5^2 : C_3):C_2):C_2$ (as 15T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![71323937370032, 70001430753280, 19747269239240, -952711121780, -1200406622100, -86339970012, 26183048420, 3053529330, -270977430, -41415725, 1334484, 273125, -2470, -855, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 855*x^13 - 2470*x^12 + 273125*x^11 + 1334484*x^10 - 41415725*x^9 - 270977430*x^8 + 3053529330*x^7 + 26183048420*x^6 - 86339970012*x^5 - 1200406622100*x^4 - 952711121780*x^3 + 19747269239240*x^2 + 70001430753280*x + 71323937370032)
 
gp: K = bnfinit(x^15 - 855*x^13 - 2470*x^12 + 273125*x^11 + 1334484*x^10 - 41415725*x^9 - 270977430*x^8 + 3053529330*x^7 + 26183048420*x^6 - 86339970012*x^5 - 1200406622100*x^4 - 952711121780*x^3 + 19747269239240*x^2 + 70001430753280*x + 71323937370032, 1)
 

Normalized defining polynomial

\( x^{15} - 855 x^{13} - 2470 x^{12} + 273125 x^{11} + 1334484 x^{10} - 41415725 x^{9} - 270977430 x^{8} + 3053529330 x^{7} + 26183048420 x^{6} - 86339970012 x^{5} - 1200406622100 x^{4} - 952711121780 x^{3} + 19747269239240 x^{2} + 70001430753280 x + 71323937370032 \)

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:  \(820758359193064981054673488281250000000000=2^{10}\cdot 5^{18}\cdot 19^{12}\cdot 37^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $622.70$
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:  $2, 5, 19, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{19} a^{5}$, $\frac{1}{38} a^{6} - \frac{1}{38} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{38} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{76} a^{8} - \frac{1}{76} a^{7} - \frac{1}{76} a^{6} - \frac{1}{76} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{76} a^{9} - \frac{1}{76} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{1444} a^{10} - \frac{1}{76} a^{6} - \frac{1}{38} a^{5}$, $\frac{1}{1444} a^{11} - \frac{1}{76} a^{7} - \frac{1}{38} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{1444} a^{12} - \frac{1}{76} a^{7} - \frac{1}{76} a^{6} - \frac{1}{76} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2888} a^{13} - \frac{1}{2888} a^{12} - \frac{1}{152} a^{9} - \frac{1}{152} a^{8} - \frac{1}{76} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{34943970012097425169416691834202004942066642831020082344} a^{14} + \frac{516664398332141440026198316259667727108240533159411}{34943970012097425169416691834202004942066642831020082344} a^{13} - \frac{162093063936623615444718854087280630069468236681863}{17471985006048712584708345917101002471033321415510041172} a^{12} + \frac{3785475872885597306322006700904006080545510021939981}{17471985006048712584708345917101002471033321415510041172} a^{11} + \frac{11627436139589045342389444422284682354031478762935927}{34943970012097425169416691834202004942066642831020082344} a^{10} - \frac{328162521526338645794802119360903438190350951711075}{96797700864535803793398038321889210365835575709196904} a^{9} - \frac{2332378611739286991103458266903712955530373780398851}{919578158213090136037281364057947498475437969237370588} a^{8} - \frac{41158298573378912521181171355437283232914604371349}{24199425216133950948349509580472302591458893927299226} a^{7} - \frac{251821515898788640596111186516779109980124803427681}{24199425216133950948349509580472302591458893927299226} a^{6} + \frac{22219321771575561108163084189715732027601208927619489}{919578158213090136037281364057947498475437969237370588} a^{5} + \frac{7069288666239749779888579575576585667692992266797861}{24199425216133950948349509580472302591458893927299226} a^{4} + \frac{6327914129333825282426038625529981235742172907424483}{24199425216133950948349509580472302591458893927299226} a^{3} + \frac{6936705913404712820060856773929690518742881645621487}{24199425216133950948349509580472302591458893927299226} a^{2} + \frac{671947021260189451613598685758497728673453364130979}{12099712608066975474174754790236151295729446963649613} a - \frac{4092460004585020634301320710151213546764463697470919}{12099712608066975474174754790236151295729446963649613}$

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:  $14$
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:  \( 607452564706000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:D_6$ (as 15T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 300
The 14 conjugacy class representatives for $((C_5^2 : C_3):C_2):C_2$
Character table for $((C_5^2 : C_3):C_2):C_2$

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 15 sibling: data not computed
Degree 25 sibling: data not computed
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{5}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$5$5.5.6.1$x^{5} + 10 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
5.10.12.10$x^{10} + 10 x^{8} + 20 x^{7} + 15 x^{6} - 5 x^{5} + 5 x^{4} + 5 x^{2} - 5 x + 7$$5$$2$$12$$D_{10}$$[3/2]_{2}^{2}$
$19$19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.10.8.2$x^{10} - 19 x^{5} + 722$$5$$2$$8$$D_5\times C_5$$[\ ]_{5}^{10}$
37Data not computed