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Properties

Label 15.1.24402933877...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{12}\cdot 3^{12}\cdot 5^{15}\cdot 7^{13}\cdot 2069^{5}$
Root discriminant $1442.66$
Ramified primes $2, 3, 5, 7, 2069$
Class number $200$ (GRH)
Class group $[5, 40]$ (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![13217245308, -2872916100, 5251526460, -1007584920, 438449760, 15925923, 27605475, 88155, 114930, 165195, 377, 7385, -370, 145, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 145*x^13 - 370*x^12 + 7385*x^11 + 377*x^10 + 165195*x^9 + 114930*x^8 + 88155*x^7 + 27605475*x^6 + 15925923*x^5 + 438449760*x^4 - 1007584920*x^3 + 5251526460*x^2 - 2872916100*x + 13217245308)
 
gp: K = bnfinit(x^15 - 5*x^14 + 145*x^13 - 370*x^12 + 7385*x^11 + 377*x^10 + 165195*x^9 + 114930*x^8 + 88155*x^7 + 27605475*x^6 + 15925923*x^5 + 438449760*x^4 - 1007584920*x^3 + 5251526460*x^2 - 2872916100*x + 13217245308, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 145 x^{13} - 370 x^{12} + 7385 x^{11} + 377 x^{10} + 165195 x^{9} + 114930 x^{8} + 88155 x^{7} + 27605475 x^{6} + 15925923 x^{5} + 438449760 x^{4} - 1007584920 x^{3} + 5251526460 x^{2} - 2872916100 x + 13217245308 \)

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:  \(-244029338779125043943806918241370375000000000000=-\,2^{12}\cdot 3^{12}\cdot 5^{15}\cdot 7^{13}\cdot 2069^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1442.66$
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:  $2, 3, 5, 7, 2069$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4}$, $\frac{1}{18} a^{9} - \frac{1}{18} a^{8} - \frac{1}{18} a^{6} - \frac{1}{9} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3}$, $\frac{1}{378} a^{10} + \frac{1}{54} a^{9} + \frac{8}{189} a^{8} + \frac{5}{54} a^{7} + \frac{19}{189} a^{6} - \frac{1}{54} a^{5} - \frac{5}{42} a^{4} - \frac{1}{3} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7}$, $\frac{1}{1134} a^{11} + \frac{1}{1134} a^{10} + \frac{8}{567} a^{9} + \frac{23}{1134} a^{8} - \frac{86}{567} a^{7} + \frac{101}{1134} a^{6} - \frac{29}{378} a^{5} + \frac{8}{63} a^{4} - \frac{2}{7} a^{3} + \frac{8}{21} a^{2} + \frac{4}{21} a - \frac{1}{7}$, $\frac{1}{1134} a^{12} + \frac{2}{81} a^{9} - \frac{19}{378} a^{8} - \frac{64}{567} a^{6} - \frac{1}{27} a^{5} - \frac{5}{126} a^{4} - \frac{1}{3} a^{3} + \frac{2}{21} a^{2} - \frac{1}{3} a + \frac{2}{7}$, $\frac{1}{3402} a^{13} + \frac{1}{3402} a^{12} + \frac{1}{3402} a^{11} - \frac{1}{3402} a^{10} + \frac{46}{1701} a^{9} + \frac{179}{3402} a^{8} - \frac{5}{189} a^{7} - \frac{65}{567} a^{6} - \frac{26}{189} a^{5} - \frac{1}{126} a^{3} + \frac{5}{21} a^{2} + \frac{1}{21} a + \frac{1}{7}$, $\frac{1}{2068891450144735182120065503340166886121297400246} a^{14} + \frac{90418215335804463904315555325884353503407127}{2068891450144735182120065503340166886121297400246} a^{13} + \frac{282279531237634639897141558899263127246431641}{1034445725072367591060032751670083443060648700123} a^{12} + \frac{22597368173455804359764297941366529925076096}{344815241690789197020010917223361147686882900041} a^{11} - \frac{2705620488991936242857295860712661578433470305}{2068891450144735182120065503340166886121297400246} a^{10} + \frac{35926346770678054631548320041525775387648146539}{2068891450144735182120065503340166886121297400246} a^{9} - \frac{7437272207439099213157778901613534345219821446}{1034445725072367591060032751670083443060648700123} a^{8} - \frac{2205758951372442895046199285726093344445006542}{16419773413847104620000519867779102270803947621} a^{7} - \frac{11072548322025733312768788556543698554555086379}{229876827793859464680007278148907431791255266694} a^{6} - \frac{3454424257643316800851679740044816194509858231}{229876827793859464680007278148907431791255266694} a^{5} - \frac{600074405903971925433087817434163369364682445}{4256978292478878975555690336090878366504727161} a^{4} + \frac{5757214895753266813876106048295935672650141813}{38312804632309910780001213024817905298542544449} a^{3} - \frac{4702499130926508557382126563052199726917756253}{12770934877436636926667071008272635099514181483} a^{2} - \frac{652723883331181516740327975631290781949234925}{4256978292478878975555690336090878366504727161} a + \frac{171359819804876893657584552641318181692406037}{4256978292478878975555690336090878366504727161}$

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

Class group and class number

$C_{5}\times C_{40}$, which has order $200$ (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:  \( 16856432353285514 \) (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

3.1.14483.1, 5.1.9724050000.9

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 R $15$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$5$5.5.5.2$x^{5} + 5 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.10.7$x^{10} + 10 x^{8} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 12$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$7$7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
7.10.9.2$x^{10} + 14$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
2069Data not computed