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Properties

Label 15.1.11244069656...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{18}\cdot 5^{28}\cdot 11^{5}\cdot 59^{5}$
Root discriminant $401.23$
Ramified primes $2, 5, 11, 59$
Class number $14$ (GRH)
Class group $[14]$ (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![4631132, 1602100, -1493480, 2047880, -570520, -165340, 55340, 34520, -14610, 4500, 629, -435, 100, 10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 10*x^13 + 100*x^12 - 435*x^11 + 629*x^10 + 4500*x^9 - 14610*x^8 + 34520*x^7 + 55340*x^6 - 165340*x^5 - 570520*x^4 + 2047880*x^3 - 1493480*x^2 + 1602100*x + 4631132)
 
gp: K = bnfinit(x^15 - 5*x^14 + 10*x^13 + 100*x^12 - 435*x^11 + 629*x^10 + 4500*x^9 - 14610*x^8 + 34520*x^7 + 55340*x^6 - 165340*x^5 - 570520*x^4 + 2047880*x^3 - 1493480*x^2 + 1602100*x + 4631132, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 10 x^{13} + 100 x^{12} - 435 x^{11} + 629 x^{10} + 4500 x^{9} - 14610 x^{8} + 34520 x^{7} + 55340 x^{6} - 165340 x^{5} - 570520 x^{4} + 2047880 x^{3} - 1493480 x^{2} + 1602100 x + 4631132 \)

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:  \(-1124406965607900390625000000000000000000=-\,2^{18}\cdot 5^{28}\cdot 11^{5}\cdot 59^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $401.23$
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:  $2, 5, 11, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{50} a^{10} + \frac{3}{10} a^{9} - \frac{1}{5} a^{7} - \frac{3}{10} a^{6} + \frac{13}{50} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a + \frac{1}{25}$, $\frac{1}{50} a^{11} - \frac{1}{2} a^{9} - \frac{1}{5} a^{8} - \frac{3}{10} a^{7} - \frac{6}{25} a^{6} - \frac{1}{2} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{1}{25} a + \frac{2}{5}$, $\frac{1}{50} a^{12} + \frac{3}{10} a^{9} - \frac{3}{10} a^{8} - \frac{6}{25} a^{7} - \frac{3}{10} a^{5} - \frac{1}{5} a^{3} + \frac{1}{25} a^{2} + \frac{2}{5} a$, $\frac{1}{550} a^{13} - \frac{1}{550} a^{12} + \frac{1}{110} a^{11} + \frac{2}{5} a^{9} - \frac{97}{550} a^{8} - \frac{13}{550} a^{7} - \frac{2}{11} a^{6} - \frac{1}{10} a^{5} - \frac{1}{5} a^{4} + \frac{6}{275} a^{3} - \frac{116}{275} a^{2} - \frac{16}{55} a + \frac{2}{5}$, $\frac{1}{196725426045984719136725888764485500} a^{14} - \frac{97030316439092370217775877691691}{196725426045984719136725888764485500} a^{13} - \frac{11917987600592303611332570488266}{4471032410136016344016497471920125} a^{12} - \frac{322351455741169535046148929264353}{98362713022992359568362944382242750} a^{11} - \frac{151770973632142799770636880068879}{17884129640544065376065989887680500} a^{10} + \frac{59645229765269681578478004966024633}{196725426045984719136725888764485500} a^{9} - \frac{23420144571944493139804857597791022}{49181356511496179784181472191121375} a^{8} - \frac{820632027908453277544360876143048}{4471032410136016344016497471920125} a^{7} - \frac{40557501899776737148344492041906799}{98362713022992359568362944382242750} a^{6} + \frac{2881968053139326841746099737200169}{8942064820272032688032994943840250} a^{5} + \frac{47612802500195527836474085213219161}{98362713022992359568362944382242750} a^{4} + \frac{17238998008378962214606033286583597}{49181356511496179784181472191121375} a^{3} + \frac{2149243461056396810803241046559453}{4471032410136016344016497471920125} a^{2} + \frac{14751785983510303925561088155075467}{49181356511496179784181472191121375} a + \frac{537331611618948122712041443447133}{4471032410136016344016497471920125}$

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

Class group and class number

$C_{14}$, which has order $14$ (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:  \( 23590463645275.965 \) (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

3.1.12980.1, 5.1.31250000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ R

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$$[\ ]_{5}^{4}$
2.10.14.5$x^{10} - 2 x^{5} - 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$5$5.5.9.2$x^{5} + 55$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.5$x^{10} + 55$$10$$1$$19$$F_5$$[9/4]_{4}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.1.1$x^{2} - 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$