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Properties

Label 15.1.19562088031...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{12}\cdot 5^{28}\cdot 19^{5}\cdot 349^{5}$
Root discriminant $659.82$
Ramified primes $2, 5, 19, 349$
Class number $21$ (GRH)
Class group $[21]$ (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![59344580, -20821100, 22609900, 1415200, 7949350, 1981059, 913895, 545195, -29340, 68715, -7051, 3985, -340, 105, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 105*x^13 - 340*x^12 + 3985*x^11 - 7051*x^10 + 68715*x^9 - 29340*x^8 + 545195*x^7 + 913895*x^6 + 1981059*x^5 + 7949350*x^4 + 1415200*x^3 + 22609900*x^2 - 20821100*x + 59344580)
 
gp: K = bnfinit(x^15 - 5*x^14 + 105*x^13 - 340*x^12 + 3985*x^11 - 7051*x^10 + 68715*x^9 - 29340*x^8 + 545195*x^7 + 913895*x^6 + 1981059*x^5 + 7949350*x^4 + 1415200*x^3 + 22609900*x^2 - 20821100*x + 59344580, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 105 x^{13} - 340 x^{12} + 3985 x^{11} - 7051 x^{10} + 68715 x^{9} - 29340 x^{8} + 545195 x^{7} + 913895 x^{6} + 1981059 x^{5} + 7949350 x^{4} + 1415200 x^{3} + 22609900 x^{2} - 20821100 x + 59344580 \)

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:  \(-1956208803104312462005615234375000000000000=-\,2^{12}\cdot 5^{28}\cdot 19^{5}\cdot 349^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $659.82$
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:  $2, 5, 19, 349$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{50} a^{10} + \frac{1}{5} a^{9} + \frac{3}{10} a^{8} + \frac{1}{10} a^{7} + \frac{1}{10} a^{6} - \frac{3}{50} a^{5} - \frac{1}{2} a^{3} - \frac{1}{5}$, $\frac{1}{50} a^{11} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{10} a^{7} + \frac{11}{25} a^{6} - \frac{2}{5} a^{5} - \frac{1}{2} a^{3} - \frac{1}{5} a$, $\frac{1}{50} a^{12} + \frac{1}{10} a^{9} - \frac{2}{5} a^{8} + \frac{11}{25} a^{7} + \frac{1}{10} a^{6} + \frac{2}{5} a^{5} - \frac{1}{2} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{50} a^{13} + \frac{1}{10} a^{9} + \frac{11}{25} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{3}{10} a^{5} - \frac{1}{5} a^{3}$, $\frac{1}{231255998402636882120305506058385507965250} a^{14} - \frac{177979276687625023940466911089822752451}{115627999201318441060152753029192753982625} a^{13} - \frac{2083708884436138040141842845891669266561}{231255998402636882120305506058385507965250} a^{12} + \frac{995010051645552489067917678830376718176}{115627999201318441060152753029192753982625} a^{11} + \frac{699694022548558779759869105744015739768}{115627999201318441060152753029192753982625} a^{10} - \frac{1565233968206386859909839809768736284164}{115627999201318441060152753029192753982625} a^{9} - \frac{32999189620350842339670559397448547701272}{115627999201318441060152753029192753982625} a^{8} + \frac{18268001177549603041558658417778788193233}{231255998402636882120305506058385507965250} a^{7} + \frac{39810885187986184635805567434088737939072}{115627999201318441060152753029192753982625} a^{6} - \frac{70352265186137053684970491067935255533033}{231255998402636882120305506058385507965250} a^{5} - \frac{6818943594630606604727200694736247204987}{46251199680527376424061101211677101593050} a^{4} - \frac{1738120088912654951819396761047954384291}{46251199680527376424061101211677101593050} a^{3} + \frac{1561589997867628624254635447548063142381}{23125599840263688212030550605838550796525} a^{2} + \frac{8137823536084587770138162221674306626333}{23125599840263688212030550605838550796525} a - \frac{3819229727935747493892985332503209612606}{23125599840263688212030550605838550796525}$

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

Class group and class number

$C_{21}$, which has order $21$ (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:  \( 158198517094902.34 \) (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

3.1.33155.1, 5.1.31250000.2

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$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}$
$5$5.5.9.5$x^{5} + 105$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.4$x^{10} + 105$$10$$1$$19$$F_5$$[9/4]_{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
349Data not computed