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Properties

Label 15.5.46431713006...9375.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,3^{6}\cdot 5^{24}\cdot 11^{12}\cdot 23^{7}$
Root discriminant $599.50$
Ramified primes $3, 5, 11, 23$
Class number $125$ (GRH)
Class group $[5, 5, 5]$ (GRH)
Galois group 15T40

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5476184846968, 7902686294580, 1660824701580, -124388611655, -50184822600, -13069638582, 2486105930, -74603760, 30190710, -6851625, 777876, -32010, -1870, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 1870*x^12 - 32010*x^11 + 777876*x^10 - 6851625*x^9 + 30190710*x^8 - 74603760*x^7 + 2486105930*x^6 - 13069638582*x^5 - 50184822600*x^4 - 124388611655*x^3 + 1660824701580*x^2 + 7902686294580*x + 5476184846968)
 
gp: K = bnfinit(x^15 - 1870*x^12 - 32010*x^11 + 777876*x^10 - 6851625*x^9 + 30190710*x^8 - 74603760*x^7 + 2486105930*x^6 - 13069638582*x^5 - 50184822600*x^4 - 124388611655*x^3 + 1660824701580*x^2 + 7902686294580*x + 5476184846968, 1)
 

Normalized defining polynomial

\( x^{15} - 1870 x^{12} - 32010 x^{11} + 777876 x^{10} - 6851625 x^{9} + 30190710 x^{8} - 74603760 x^{7} + 2486105930 x^{6} - 13069638582 x^{5} - 50184822600 x^{4} - 124388611655 x^{3} + 1660824701580 x^{2} + 7902686294580 x + 5476184846968 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-464317130069216763190044343471527099609375=-\,3^{6}\cdot 5^{24}\cdot 11^{12}\cdot 23^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $599.50$
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:  $3, 5, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11} a^{5}$, $\frac{1}{11} a^{6}$, $\frac{1}{22} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{66} a^{8} - \frac{1}{66} a^{7} + \frac{1}{33} a^{6} + \frac{1}{66} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{66} a^{9} + \frac{1}{66} a^{7} - \frac{1}{22} a^{6} - \frac{1}{33} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{726} a^{10} - \frac{1}{66} a^{7} - \frac{1}{33} a^{6} + \frac{1}{33} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{726} a^{11} - \frac{1}{33} a^{6} - \frac{1}{33} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{726} a^{12} + \frac{1}{66} a^{7} - \frac{1}{33} a^{6} - \frac{1}{33} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{50094} a^{13} - \frac{5}{50094} a^{12} - \frac{1}{1518} a^{11} + \frac{17}{25047} a^{10} - \frac{2}{2277} a^{9} - \frac{4}{759} a^{8} + \frac{32}{2277} a^{7} + \frac{86}{2277} a^{6} + \frac{6}{253} a^{5} - \frac{15}{46} a^{4} - \frac{53}{138} a^{3} + \frac{17}{138} a^{2} + \frac{68}{207} a - \frac{52}{207}$, $\frac{1}{574915675750128710278022897305448884047788219911678812108496443609012} a^{14} + \frac{546464433649414579740235837063868059110596452107622768645692851}{143728918937532177569505724326362221011947054977919703027124110902253} a^{13} - \frac{626866717666865109425484358137682737232244708217226141737097570}{13066265357957470688136884029669292819267914088901791184284010082023} a^{12} - \frac{151788519967669599389198404607608854208747412765953792832717927047}{287457837875064355139011448652724442023894109955839406054248221804506} a^{11} - \frac{51415155447219734155492795758688977564088290127059008445485528105}{287457837875064355139011448652724442023894109955839406054248221804506} a^{10} - \frac{6157699225445112301202770556522188475329687379102484057000966565}{13066265357957470688136884029669292819267914088901791184284010082023} a^{9} - \frac{3966478753559051743796409726634004826015123498712498093106757629}{7466437347404268964649648016953881611010236622229594962448005761156} a^{8} - \frac{251542841646225229648703866060867487769373775933969882930785353653}{26132530715914941376273768059338585638535828177803582368568020164046} a^{7} - \frac{338931845121499631532629114524508957358739919089574271643081866632}{13066265357957470688136884029669292819267914088901791184284010082023} a^{6} + \frac{42055261910054952874350868054079790241932549836681333256538515367}{8710843571971647125424589353112861879511942725934527456189340054682} a^{5} - \frac{43551981661212564390993571065310769998806623023679634806603766399}{263964956726413549255290586457965511500361900785894771399676971354} a^{4} + \frac{10055163553112008483595179736142308404309287861207310314270964232}{131982478363206774627645293228982755750180950392947385699838485677} a^{3} + \frac{2187290235653635224804342868196243286605135095653272073841945329421}{4751369221075443886595230556243379207006514214146105885194185484372} a^{2} + \frac{33313699776656138922354280244867173502644609928804508891593261321}{1187842305268860971648807639060844801751628553536526471298546371093} a + \frac{535278114812998967383787591472296784382295491984733535463352913709}{1187842305268860971648807639060844801751628553536526471298546371093}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{5}$, which has order $125$ (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:  $9$
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:  \( 20411798198200 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T40:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1500
The 40 conjugacy class representatives for [5^3:2]S(3)
Character table for [5^3:2]S(3) is not computed

Intermediate fields

3.1.23.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 15 siblings: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.5.8.1$x^{5} - 5 x^{4} + 105$$5$$1$$8$$C_5$$[2]$
5.10.16.8$x^{10} + 40 x^{9} + 10 x^{8} + 70 x^{7} + 15 x^{6} + 20 x^{5} + 5 x^{4} + 80 x^{3} - 20 x^{2} + 90 x + 7$$5$$2$$16$$C_{10}$$[2]^{2}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.2$x^{10} + 143 x^{5} + 5929$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$