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Properties

Label 15.1.21596876454...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{18}\cdot 5^{15}\cdot 11^{13}\cdot 379^{5}$
Root discriminant $664.19$
Ramified primes $2, 5, 11, 379$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![93869216, -74460160, 50225840, -22766960, 20480, 3131424, 954040, -862280, 138610, 21460, -10957, 405, 400, -50, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 50*x^13 + 400*x^12 + 405*x^11 - 10957*x^10 + 21460*x^9 + 138610*x^8 - 862280*x^7 + 954040*x^6 + 3131424*x^5 + 20480*x^4 - 22766960*x^3 + 50225840*x^2 - 74460160*x + 93869216)
 
gp: K = bnfinit(x^15 - 5*x^14 - 50*x^13 + 400*x^12 + 405*x^11 - 10957*x^10 + 21460*x^9 + 138610*x^8 - 862280*x^7 + 954040*x^6 + 3131424*x^5 + 20480*x^4 - 22766960*x^3 + 50225840*x^2 - 74460160*x + 93869216, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 50 x^{13} + 400 x^{12} + 405 x^{11} - 10957 x^{10} + 21460 x^{9} + 138610 x^{8} - 862280 x^{7} + 954040 x^{6} + 3131424 x^{5} + 20480 x^{4} - 22766960 x^{3} + 50225840 x^{2} - 74460160 x + 93869216 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2159687645407390587226247752000000000000000=-\,2^{18}\cdot 5^{15}\cdot 11^{13}\cdot 379^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $664.19$
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:  $2, 5, 11, 379$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5}$, $\frac{1}{44} a^{10} - \frac{5}{44} a^{8} - \frac{1}{44} a^{6} + \frac{1}{44} a^{4} + \frac{4}{11} a^{2} - \frac{3}{11}$, $\frac{1}{88} a^{11} - \frac{1}{88} a^{10} + \frac{3}{44} a^{9} - \frac{3}{44} a^{8} + \frac{21}{88} a^{7} - \frac{21}{88} a^{6} + \frac{3}{22} a^{5} - \frac{3}{22} a^{4} + \frac{2}{11} a^{3} - \frac{2}{11} a^{2} + \frac{4}{11} a - \frac{4}{11}$, $\frac{1}{176} a^{12} - \frac{1}{176} a^{11} - \frac{1}{88} a^{10} + \frac{1}{11} a^{9} + \frac{17}{176} a^{8} - \frac{21}{176} a^{7} + \frac{5}{44} a^{6} + \frac{5}{88} a^{5} - \frac{9}{44} a^{4} + \frac{7}{44} a^{3} + \frac{5}{11} a^{2} + \frac{7}{22} a - \frac{5}{11}$, $\frac{1}{176} a^{13} - \frac{1}{176} a^{11} + \frac{1}{176} a^{9} + \frac{41}{176} a^{7} + \frac{21}{88} a^{5} - \frac{9}{44} a^{3} - \frac{1}{2} a^{2} + \frac{5}{22} a$, $\frac{1}{966986416121815752983865804066930971552} a^{14} + \frac{2302880584313025474533378789165684129}{966986416121815752983865804066930971552} a^{13} + \frac{558574311188738140734297890265987791}{483493208060907876491932902033465485776} a^{12} - \frac{2188120320986196626779264252284463403}{483493208060907876491932902033465485776} a^{11} - \frac{4999605765725525104192928779985530183}{966986416121815752983865804066930971552} a^{10} + \frac{70940824865507355713863565455232972925}{966986416121815752983865804066930971552} a^{9} + \frac{14722656519469974608003589833940751881}{241746604030453938245966451016732742888} a^{8} - \frac{352217620403710064900727783851801847}{10988482001384269920271202318942397404} a^{7} - \frac{58473478058690053779048239900330213549}{241746604030453938245966451016732742888} a^{6} - \frac{26570128444862511523625344592695558791}{120873302015226969122983225508366371444} a^{5} - \frac{13677207979017752620253726898261551439}{120873302015226969122983225508366371444} a^{4} - \frac{15537492166307225675419015067174859123}{60436651007613484561491612754183185722} a^{3} - \frac{6290354510580610920924749000141096735}{30218325503806742280745806377091592861} a^{2} + \frac{24317811924880246446680859794233981769}{60436651007613484561491612754183185722} a - \frac{5020593285052352286090672569938318539}{30218325503806742280745806377091592861}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  $7$
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:  \( 193308478043254.28 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.16676.1, 5.1.732050000.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$5$5.15.15.40$x^{15} + 10 x^{14} + 5 x^{13} + 20 x^{12} + 20 x^{11} + 17 x^{10} + 10 x^{9} + 15 x^{8} + 12 x^{5} + 15 x^{3} + 10 x^{2} + 20 x + 17$$5$$3$$15$$F_5\times C_3$$[5/4]_{4}^{3}$
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$
379Data not computed