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Properties

Label 15.1.48132773690...0000.1
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![25424027623886288, -2845372937103000, 241484266064340, -16823896467110, -7105193819530, 689232153107, -7268082900, 5895045565, 336699800, 29553325, -4322064, -43740, -3120, 155, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 155*x^13 - 3120*x^12 - 43740*x^11 - 4322064*x^10 + 29553325*x^9 + 336699800*x^8 + 5895045565*x^7 - 7268082900*x^6 + 689232153107*x^5 - 7105193819530*x^4 - 16823896467110*x^3 + 241484266064340*x^2 - 2845372937103000*x + 25424027623886288)
 
gp: K = bnfinit(x^15 + 155*x^13 - 3120*x^12 - 43740*x^11 - 4322064*x^10 + 29553325*x^9 + 336699800*x^8 + 5895045565*x^7 - 7268082900*x^6 + 689232153107*x^5 - 7105193819530*x^4 - 16823896467110*x^3 + 241484266064340*x^2 - 2845372937103000*x + 25424027623886288, 1)
 

Normalized defining polynomial

\( x^{15} + 155 x^{13} - 3120 x^{12} - 43740 x^{11} - 4322064 x^{10} + 29553325 x^{9} + 336699800 x^{8} + 5895045565 x^{7} - 7268082900 x^{6} + 689232153107 x^{5} - 7105193819530 x^{4} - 16823896467110 x^{3} + 241484266064340 x^{2} - 2845372937103000 x + 25424027623886288 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{424} a^{13} + \frac{1}{53} a^{12} + \frac{47}{424} a^{11} + \frac{15}{53} a^{10} + \frac{2}{53} a^{9} + \frac{4}{53} a^{8} + \frac{149}{424} a^{7} + \frac{7}{53} a^{6} - \frac{15}{424} a^{5} + \frac{5}{106} a^{4} + \frac{3}{8} a^{3} - \frac{73}{212} a^{2} + \frac{15}{212} a + \frac{9}{106}$, $\frac{1}{771704141712442265672300524691001320632541507784577763979500273364688672494400164718600436415009181069216} a^{14} - \frac{86075953606646148989491638113810625875633785693481120936823826844460925990204011395622299898541371975}{385852070856221132836150262345500660316270753892288881989750136682344336247200082359300218207504590534608} a^{13} + \frac{125077608182346363875670768623871384559171415004313521587539036904117629809514620154023043568612027798751}{771704141712442265672300524691001320632541507784577763979500273364688672494400164718600436415009181069216} a^{12} + \frac{186070000445047595486057496704543235009462390991050018177518363451763699781707291417183493127416547689743}{385852070856221132836150262345500660316270753892288881989750136682344336247200082359300218207504590534608} a^{11} - \frac{3590565322206642997129977969416892213867686625181825731450959617257624087933238315869923029681511256240}{24115754428513820802259391396593791269766922118268055124359383542646521015450005147456263637969036908413} a^{10} + \frac{14059985158500735466424921761718686419506780656000711099346816092508159983028604195415742237856285800945}{48231508857027641604518782793187582539533844236536110248718767085293042030900010294912527275938073816826} a^{9} - \frac{5232934328284733087381841164444614173731919179602755865151816699372677146671322244539269586749434027123}{771704141712442265672300524691001320632541507784577763979500273364688672494400164718600436415009181069216} a^{8} - \frac{171480400745757997548847628703771960434331137800055143317907634183672230480880990168893341906607123312479}{385852070856221132836150262345500660316270753892288881989750136682344336247200082359300218207504590534608} a^{7} - \frac{82045111467235317624371186632400029935615443979678746853022251254223758439225256337335119413564814098143}{771704141712442265672300524691001320632541507784577763979500273364688672494400164718600436415009181069216} a^{6} - \frac{71378505943132799009138643125047018125976587368859895060468905966724866766446486752180914639035556189089}{385852070856221132836150262345500660316270753892288881989750136682344336247200082359300218207504590534608} a^{5} + \frac{203407461199412531568001079283725380217127043255480410356910821644448408909562812766498507904233030563887}{771704141712442265672300524691001320632541507784577763979500273364688672494400164718600436415009181069216} a^{4} + \frac{59667906594739088859135202483913775513638728207134821044189228214137650071517411011514448169333651303105}{192926035428110566418075131172750330158135376946144440994875068341172168123600041179650109103752295267304} a^{3} + \frac{100033432948641325679052036075242985345663606855370736588247673388138883794750950578179952710383618944273}{385852070856221132836150262345500660316270753892288881989750136682344336247200082359300218207504590534608} a^{2} + \frac{36878655143608514574747055385298736573086547308643167411171154896659711256734436976455975296098391432619}{96463017714055283209037565586375165079067688473072220497437534170586084061800020589825054551876147633652} a + \frac{25180025418102298756576877852086098522529149834742441824770997209292518006810751720522374401364367558967}{96463017714055283209037565586375165079067688473072220497437534170586084061800020589825054551876147633652}$

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:  \( 17603642299200000 \) (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 ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{5}$

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}$