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Properties

Label 15.1.48132773690...0000.2
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{21}\cdot 5^{18}\cdot 11^{12}\cdot 61^{8}$
Root discriminant $1110.44$
Ramified primes $2, 5, 11, 61$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group 15T31

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8093159542525366, -908213995898190, -302495276454075, 92007754171735, 9883081080065, -1546253890847, -91266360040, 11648990810, 112654015, -49110085, 1097623, 178955, -1345, -615, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 615*x^13 - 1345*x^12 + 178955*x^11 + 1097623*x^10 - 49110085*x^9 + 112654015*x^8 + 11648990810*x^7 - 91266360040*x^6 - 1546253890847*x^5 + 9883081080065*x^4 + 92007754171735*x^3 - 302495276454075*x^2 - 908213995898190*x + 8093159542525366)
 
gp: K = bnfinit(x^15 - 5*x^14 - 615*x^13 - 1345*x^12 + 178955*x^11 + 1097623*x^10 - 49110085*x^9 + 112654015*x^8 + 11648990810*x^7 - 91266360040*x^6 - 1546253890847*x^5 + 9883081080065*x^4 + 92007754171735*x^3 - 302495276454075*x^2 - 908213995898190*x + 8093159542525366, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 615 x^{13} - 1345 x^{12} + 178955 x^{11} + 1097623 x^{10} - 49110085 x^{9} + 112654015 x^{8} + 11648990810 x^{7} - 91266360040 x^{6} - 1546253890847 x^{5} + 9883081080065 x^{4} + 92007754171735 x^{3} - 302495276454075 x^{2} - 908213995898190 x + 8093159542525366 \)

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:  \(-4813277369084810191333564808000000000000000000=-\,2^{21}\cdot 5^{18}\cdot 11^{12}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1110.44$
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:  $2, 5, 11, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6513443993164438524114321045356501189774128151098661952477764624209994219215324596960911792575976} a^{14} - \frac{583236690576378742549179541328765317560184584473221016546179998704893293293519587708676211024243}{3256721996582219262057160522678250594887064075549330976238882312104997109607662298480455896287988} a^{13} - \frac{602812460048051483327957151173979719051620010237464138469678674487621892062600537694333533476617}{6513443993164438524114321045356501189774128151098661952477764624209994219215324596960911792575976} a^{12} - \frac{13563371714624339113466395294178300178678583222985508537299857932645618856645234176701639119046}{62629269165042678116483856205350972978597386068256364927670813694326867492455044201547228774769} a^{11} - \frac{2935198413267536636500375941641559734481246498880158018420231656685385845793067993472097439877013}{6513443993164438524114321045356501189774128151098661952477764624209994219215324596960911792575976} a^{10} + \frac{16935474202776690268891043217491968293656223222731769063636030261527735765648191342050567734377}{125258538330085356232967712410701945957194772136512729855341627388653734984910088403094457549538} a^{9} + \frac{436221863661988814030413448662680943472268628846326102980003877450526440092049487963866942120095}{6513443993164438524114321045356501189774128151098661952477764624209994219215324596960911792575976} a^{8} - \frac{404068958266819090381257502057693938638603647809092682453761553925513553882580023756265316080739}{814180499145554815514290130669562648721766018887332744059720578026249277401915574620113974071997} a^{7} - \frac{1334167704889444378423050277498113217310561351258294853597663559417766304915679180688777387410523}{3256721996582219262057160522678250594887064075549330976238882312104997109607662298480455896287988} a^{6} - \frac{154177083067426146567556613748867747472786890902415970832185682350710213827857971874912545248425}{3256721996582219262057160522678250594887064075549330976238882312104997109607662298480455896287988} a^{5} - \frac{1791063794777651769978330820063662528324736750532544904781593732061060003272521130727317213835269}{6513443993164438524114321045356501189774128151098661952477764624209994219215324596960911792575976} a^{4} + \frac{1069574385686242442571600704949491531543092626697210015498416439872038903721082472189169695918367}{3256721996582219262057160522678250594887064075549330976238882312104997109607662298480455896287988} a^{3} - \frac{2508253385321251481698443078253542743384299752457385962834743295147250736115818145059283205343423}{6513443993164438524114321045356501189774128151098661952477764624209994219215324596960911792575976} a^{2} - \frac{603290195713207044042926830511029248325418426070653444846962807801334262966309444608805725596915}{1628360998291109631028580261339125297443532037774665488119441156052498554803831149240227948143994} a - \frac{1470895949561606348874035559514915187734303911263876796620492135278024146046088721832558081066609}{3256721996582219262057160522678250594887064075549330976238882312104997109607662298480455896287988}$

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:  \( 21001356499000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T31:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 750
The 35 conjugacy class representatives for 1/2[5^3:2]S(3)
Character table for 1/2[5^3:2]S(3) is not computed

Intermediate fields

3.1.200.1

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 R $15$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{5}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
5Data not computed
11Data not computed
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.10.8.5$x^{10} - 61 x^{5} + 59536$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$