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Properties

Label 15.1.20294341129...4375.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,5^{24}\cdot 23^{7}$
Root discriminant $56.73$
Ramified primes $5, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_5^2 : C_3):C_2$ (as 15T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1504832, 1167680, 1319840, 523280, 235700, 12368, -12895, -9565, -405, 1100, 542, 95, -45, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^13 - 45*x^12 + 95*x^11 + 542*x^10 + 1100*x^9 - 405*x^8 - 9565*x^7 - 12895*x^6 + 12368*x^5 + 235700*x^4 + 523280*x^3 + 1319840*x^2 + 1167680*x + 1504832)
 
gp: K = bnfinit(x^15 - 5*x^13 - 45*x^12 + 95*x^11 + 542*x^10 + 1100*x^9 - 405*x^8 - 9565*x^7 - 12895*x^6 + 12368*x^5 + 235700*x^4 + 523280*x^3 + 1319840*x^2 + 1167680*x + 1504832, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{13} - 45 x^{12} + 95 x^{11} + 542 x^{10} + 1100 x^{9} - 405 x^{8} - 9565 x^{7} - 12895 x^{6} + 12368 x^{5} + 235700 x^{4} + 523280 x^{3} + 1319840 x^{2} + 1167680 x + 1504832 \)

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:  \(-202943411290645599365234375=-\,5^{24}\cdot 23^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.73$
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:  $5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{10} + \frac{1}{6} a^{7} - \frac{1}{4} a^{6} + \frac{1}{6} a^{5} - \frac{1}{4} a^{4} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{8} - \frac{1}{4} a^{7} + \frac{1}{6} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{6} a^{3} - \frac{5}{12} a^{2} + \frac{1}{6} a$, $\frac{1}{96} a^{12} + \frac{1}{48} a^{11} - \frac{1}{96} a^{10} - \frac{7}{96} a^{9} + \frac{1}{96} a^{8} + \frac{1}{24} a^{6} - \frac{5}{96} a^{5} - \frac{23}{96} a^{4} + \frac{9}{32} a^{3} - \frac{3}{16} a^{2} - \frac{1}{2} a + \frac{5}{12}$, $\frac{1}{192} a^{13} + \frac{1}{64} a^{11} - \frac{5}{192} a^{10} - \frac{3}{64} a^{9} + \frac{7}{96} a^{8} - \frac{5}{48} a^{7} + \frac{1}{64} a^{6} - \frac{13}{192} a^{5} + \frac{25}{192} a^{4} + \frac{5}{12} a^{3} - \frac{7}{48} a^{2} - \frac{11}{24} a - \frac{5}{12}$, $\frac{1}{8325914598136710761841355130496} a^{14} + \frac{8738662633968643567023897107}{4162957299068355380920677565248} a^{13} + \frac{18698101520348116576885876931}{8325914598136710761841355130496} a^{12} - \frac{26126027522439604117870019155}{8325914598136710761841355130496} a^{11} + \frac{56272159320057600016664307625}{8325914598136710761841355130496} a^{10} - \frac{24981816038039422244672989019}{346913108255696281743389797104} a^{9} + \frac{3428753140604814130129106692}{65046207797943052826885586957} a^{8} - \frac{1023245397441856586868580958197}{8325914598136710761841355130496} a^{7} + \frac{660506539032227794096451667925}{8325914598136710761841355130496} a^{6} - \frac{541263929532522815238491112071}{2775304866045570253947118376832} a^{5} + \frac{201896196752123297496693247867}{4162957299068355380920677565248} a^{4} + \frac{77965728917715479995341682547}{189225331775834335496394434784} a^{3} - \frac{10074446403853194433598892965}{86728277063924070435847449276} a^{2} + \frac{42144653439019908362921589511}{260184831191772211307542347828} a - \frac{93975527832714931385326871455}{260184831191772211307542347828}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 26468773.865 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:S_3$ (as 15T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 150
The 13 conjugacy class representatives for $(C_5^2 : C_3):C_2$
Character table for $(C_5^2 : C_3):C_2$

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 sibling: data not computed
Degree 25 sibling: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$5$5.5.8.6$x^{5} + 5 x^{4} + 5$$5$$1$$8$$D_{5}$$[2]^{2}$
5.10.16.24$x^{10} + 30 x^{9} + 25 x^{8} + 10 x^{5} + 525 x^{4} + 25$$5$$2$$16$$D_5\times C_5$$[2, 2]^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.10.5.2$x^{10} - 279841 x^{2} + 12872686$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$