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Properties

Label 15.1.30904367908...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 3^{13}\cdot 5^{17}\cdot 13^{13}$
Root discriminant $429.21$
Ramified primes $2, 3, 5, 13$
Class number $20$ (GRH)
Class group $[2, 10]$ (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![5856994189, 13423647535, 6121946705, 203969475, -125865275, 132709223, 28332665, -3692325, 231705, 277655, -16633, -1235, 705, -25, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 25*x^13 + 705*x^12 - 1235*x^11 - 16633*x^10 + 277655*x^9 + 231705*x^8 - 3692325*x^7 + 28332665*x^6 + 132709223*x^5 - 125865275*x^4 + 203969475*x^3 + 6121946705*x^2 + 13423647535*x + 5856994189)
 
gp: K = bnfinit(x^15 - 5*x^14 - 25*x^13 + 705*x^12 - 1235*x^11 - 16633*x^10 + 277655*x^9 + 231705*x^8 - 3692325*x^7 + 28332665*x^6 + 132709223*x^5 - 125865275*x^4 + 203969475*x^3 + 6121946705*x^2 + 13423647535*x + 5856994189, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 25 x^{13} + 705 x^{12} - 1235 x^{11} - 16633 x^{10} + 277655 x^{9} + 231705 x^{8} - 3692325 x^{7} + 28332665 x^{6} + 132709223 x^{5} - 125865275 x^{4} + 203969475 x^{3} + 6121946705 x^{2} + 13423647535 x + 5856994189 \)

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:  \(-3090436790831875710201600000000000000000=-\,2^{23}\cdot 3^{13}\cdot 5^{17}\cdot 13^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $429.21$
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:  $2, 3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{858} a^{10} + \frac{21}{143} a^{9} + \frac{29}{78} a^{8} - \frac{40}{143} a^{7} + \frac{53}{429} a^{6} + \frac{35}{143} a^{5} - \frac{68}{429} a^{4} - \frac{2}{13} a^{3} - \frac{397}{858} a^{2} - \frac{15}{143} a - \frac{217}{858}$, $\frac{1}{858} a^{11} - \frac{113}{858} a^{9} - \frac{18}{143} a^{8} + \frac{158}{429} a^{7} - \frac{46}{143} a^{6} + \frac{1}{429} a^{5} - \frac{2}{11} a^{4} - \frac{67}{858} a^{3} + \frac{28}{143} a^{2} - \frac{31}{858} a - \frac{19}{143}$, $\frac{1}{858} a^{12} + \frac{67}{143} a^{9} + \frac{109}{286} a^{8} + \frac{10}{143} a^{7} - \frac{16}{429} a^{6} + \frac{68}{143} a^{5} + \frac{3}{286} a^{4} - \frac{27}{143} a^{3} - \frac{46}{143} a^{2} + \frac{2}{143} a + \frac{361}{858}$, $\frac{1}{24300276} a^{13} - \frac{647}{1157156} a^{12} + \frac{1759}{6075069} a^{11} - \frac{4}{21021} a^{10} - \frac{6591407}{24300276} a^{9} + \frac{7404125}{24300276} a^{8} + \frac{1630424}{6075069} a^{7} + \frac{36242}{6075069} a^{6} + \frac{839305}{2209116} a^{5} + \frac{694721}{2209116} a^{4} - \frac{3735365}{12150138} a^{3} - \frac{2565469}{12150138} a^{2} + \frac{46225}{1157156} a - \frac{4363259}{24300276}$, $\frac{1}{137854013674700658301664113613427653358365989843284} a^{14} - \frac{2262880731484295659899568012456283356442659}{137854013674700658301664113613427653358365989843284} a^{13} + \frac{10794828428694254383756114419271035291985385907}{22975668945783443050277352268904608893060998307214} a^{12} - \frac{15353303737541835518268512852097873124306921387}{68927006837350329150832056806713826679182994921642} a^{11} + \frac{20811967830694073128271285635895747901560799383}{137854013674700658301664113613427653358365989843284} a^{10} - \frac{2537338447513530615058430552153454132569836298329}{8109059627923568135392006683142803138727411167252} a^{9} + \frac{419284121617101254951358318871497654891090791252}{1641119210413103075019810876350329206647214164801} a^{8} + \frac{3917171819702653019991338198945950766124811500674}{11487834472891721525138676134452304446530499153607} a^{7} - \frac{21785639760285120093314476402255877822135067259969}{137854013674700658301664113613427653358365989843284} a^{6} - \frac{15293964681714219356370047175265753385348409073105}{137854013674700658301664113613427653358365989843284} a^{5} + \frac{538313542689376616061412096458812121539656934098}{34463503418675164575416028403356913339591497460821} a^{4} - \frac{712962288594981177286571234295627809230345935065}{3133045765334105870492366218486992121781045223711} a^{3} + \frac{63204456554607863319511694935407204642063814353755}{137854013674700658301664113613427653358365989843284} a^{2} - \frac{2967872016114442709526458623970032664966983514641}{8109059627923568135392006683142803138727411167252} a - \frac{12991908892504559948542399399706929154120001552410}{34463503418675164575416028403356913339591497460821}$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:  \( 29832795961851.938 \) (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

3.1.7800.1, 5.1.115672050000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{5}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.19.1$x^{10} - 2$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.15.17.3$x^{15} - 5 x^{13} + 10 x^{12} + 10 x^{11} + 10 x^{9} - 5 x^{8} - 10 x^{6} + 10 x^{5} + 5 x^{3} + 10$$15$$1$$17$$F_5 \times S_3$$[5/4]_{12}^{2}$
$13$13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
13.10.9.2$x^{10} + 26$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$