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Properties

Label 15.3.64846221839...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{10}\cdot 3^{13}\cdot 5^{10}\cdot 19^{12}\cdot 179^{5}$
Root discriminant $714.70$
Ramified primes $2, 3, 5, 19, 179$
Class number $25$ (GRH)
Class group $[5, 5]$ (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![-398276804, -163462160, 409630610, -195801075, 21244810, 22965638, -9133825, -763005, 926130, -69745, -35470, 4765, 675, -115, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 115*x^13 + 675*x^12 + 4765*x^11 - 35470*x^10 - 69745*x^9 + 926130*x^8 - 763005*x^7 - 9133825*x^6 + 22965638*x^5 + 21244810*x^4 - 195801075*x^3 + 409630610*x^2 - 163462160*x - 398276804)
 
gp: K = bnfinit(x^15 - 5*x^14 - 115*x^13 + 675*x^12 + 4765*x^11 - 35470*x^10 - 69745*x^9 + 926130*x^8 - 763005*x^7 - 9133825*x^6 + 22965638*x^5 + 21244810*x^4 - 195801075*x^3 + 409630610*x^2 - 163462160*x - 398276804, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 115 x^{13} + 675 x^{12} + 4765 x^{11} - 35470 x^{10} - 69745 x^{9} + 926130 x^{8} - 763005 x^{7} - 9133825 x^{6} + 22965638 x^{5} + 21244810 x^{4} - 195801075 x^{3} + 409630610 x^{2} - 163462160 x - 398276804 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6484622183937592099059160665866970000000000=2^{10}\cdot 3^{13}\cdot 5^{10}\cdot 19^{12}\cdot 179^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $714.70$
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:  $2, 3, 5, 19, 179$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{30} a^{10} + \frac{1}{30} a^{8} - \frac{2}{5} a^{7} - \frac{4}{15} a^{6} + \frac{2}{5} a^{5} - \frac{2}{15} a^{4} - \frac{7}{30} a^{2} - \frac{1}{5} a + \frac{4}{15}$, $\frac{1}{30} a^{11} + \frac{1}{30} a^{9} - \frac{1}{15} a^{7} + \frac{1}{5} a^{6} - \frac{2}{15} a^{5} + \frac{2}{5} a^{4} - \frac{7}{30} a^{3} - \frac{2}{15} a + \frac{2}{5}$, $\frac{1}{150} a^{12} + \frac{1}{150} a^{11} - \frac{1}{150} a^{10} - \frac{1}{30} a^{9} - \frac{1}{15} a^{8} + \frac{11}{75} a^{7} + \frac{7}{25} a^{6} - \frac{11}{75} a^{5} - \frac{1}{30} a^{4} + \frac{13}{30} a^{3} - \frac{19}{75} a^{2} + \frac{31}{75} a + \frac{19}{75}$, $\frac{1}{150} a^{13} - \frac{1}{75} a^{11} + \frac{1}{150} a^{10} - \frac{1}{30} a^{9} + \frac{7}{150} a^{8} + \frac{2}{15} a^{7} - \frac{7}{75} a^{6} - \frac{73}{150} a^{5} + \frac{2}{15} a^{4} + \frac{47}{150} a^{3} - \frac{1}{6} a^{2} - \frac{4}{25} a - \frac{14}{75}$, $\frac{1}{1056073620368122576900278588370781942645754450} a^{14} - \frac{1433966458439741938958242173219171847086641}{528036810184061288450139294185390971322877225} a^{13} + \frac{663726030356441216138144159500032976694417}{211214724073624515380055717674156388529150890} a^{12} + \frac{4680312892462174449812835230706679222273946}{528036810184061288450139294185390971322877225} a^{11} - \frac{3624412899999134433247861091019103994641789}{1056073620368122576900278588370781942645754450} a^{10} - \frac{37405335770263880378591893996031938998482509}{528036810184061288450139294185390971322877225} a^{9} - \frac{31841375271161135977453898268653815177224403}{352024540122707525633426196123593980881918150} a^{8} - \frac{5979773612228663607726850221318552592569319}{35202454012270752563342619612359398088191815} a^{7} + \frac{90610733163696993061055812064975022848601113}{352024540122707525633426196123593980881918150} a^{6} - \frac{9563681983732455526972623141457548179967344}{528036810184061288450139294185390971322877225} a^{5} - \frac{173478932601423340218943220454255458619560119}{528036810184061288450139294185390971322877225} a^{4} + \frac{117088279931031640619225519448968110477022338}{528036810184061288450139294185390971322877225} a^{3} + \frac{66102179231573069674046845665535741541727553}{211214724073624515380055717674156388529150890} a^{2} - \frac{68184696127696396206619291294866989769118213}{528036810184061288450139294185390971322877225} a + \frac{121181601961519006099430977602037779842065381}{528036810184061288450139294185390971322877225}$

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:  $8$
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:  \( 797493322982159.4 \) (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

3.3.10740.1, 5.1.1319500125.1

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 R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
$179$$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.1.1$x^{2} - 179$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.4.2.1$x^{4} + 2327 x^{2} + 1570009$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
179.4.2.1$x^{4} + 2327 x^{2} + 1570009$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$