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Properties

Label 15.1.11522759010...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 3^{13}\cdot 5^{17}\cdot 41^{13}$
Root discriminant $636.95$
Ramified primes $2, 3, 5, 41$
Class number $75$ (GRH)
Class group $[5, 15]$ (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![-48460084002, 1653437610, -21861208260, 3220463205, -1841961870, -14726448, 41596335, -18871785, -340470, 62805, -8284, 1265, -265, -5, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 5*x^13 - 265*x^12 + 1265*x^11 - 8284*x^10 + 62805*x^9 - 340470*x^8 - 18871785*x^7 + 41596335*x^6 - 14726448*x^5 - 1841961870*x^4 + 3220463205*x^3 - 21861208260*x^2 + 1653437610*x - 48460084002)
 
gp: K = bnfinit(x^15 - 5*x^14 - 5*x^13 - 265*x^12 + 1265*x^11 - 8284*x^10 + 62805*x^9 - 340470*x^8 - 18871785*x^7 + 41596335*x^6 - 14726448*x^5 - 1841961870*x^4 + 3220463205*x^3 - 21861208260*x^2 + 1653437610*x - 48460084002, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 5 x^{13} - 265 x^{12} + 1265 x^{11} - 8284 x^{10} + 62805 x^{9} - 340470 x^{8} - 18871785 x^{7} + 41596335 x^{6} - 14726448 x^{5} - 1841961870 x^{4} + 3220463205 x^{3} - 21861208260 x^{2} + 1653437610 x - 48460084002 \)

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:  \(-1152275901087640214484331783593750000000000=-\,2^{10}\cdot 3^{13}\cdot 5^{17}\cdot 41^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $636.95$
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:  $2, 3, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{15} a^{5} + \frac{1}{3} a^{2} - \frac{1}{5}$, $\frac{1}{45} a^{6} + \frac{1}{9} a^{3} - \frac{1}{3} a^{2} - \frac{2}{5} a$, $\frac{1}{45} a^{7} + \frac{1}{9} a^{4} - \frac{1}{3} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{135} a^{8} - \frac{1}{135} a^{7} - \frac{4}{135} a^{5} - \frac{4}{27} a^{4} - \frac{1}{45} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{7425} a^{9} - \frac{17}{7425} a^{8} + \frac{64}{7425} a^{7} - \frac{13}{7425} a^{6} - \frac{14}{675} a^{5} + \frac{1187}{7425} a^{4} - \frac{218}{2475} a^{3} - \frac{298}{825} a^{2} - \frac{103}{275} a - \frac{9}{25}$, $\frac{1}{913275} a^{10} + \frac{37}{913275} a^{9} - \frac{854}{913275} a^{8} - \frac{737}{83025} a^{7} + \frac{4094}{913275} a^{6} + \frac{17621}{913275} a^{5} + \frac{44798}{304425} a^{4} + \frac{22178}{101475} a^{3} - \frac{722}{3075} a^{2} - \frac{3262}{11275} a + \frac{488}{1025}$, $\frac{1}{2739825} a^{11} + \frac{1}{2739825} a^{10} + \frac{28}{2739825} a^{9} + \frac{5294}{2739825} a^{8} - \frac{29143}{2739825} a^{7} + \frac{4822}{547965} a^{6} - \frac{2941}{913275} a^{5} - \frac{1019}{33825} a^{4} - \frac{48086}{101475} a^{3} - \frac{12836}{33825} a^{2} - \frac{16354}{33825} a + \frac{67}{205}$, $\frac{1}{41097375} a^{12} + \frac{4}{41097375} a^{11} + \frac{4}{41097375} a^{10} - \frac{61}{1643895} a^{9} - \frac{116}{149445} a^{8} + \frac{41549}{41097375} a^{7} + \frac{9907}{13699125} a^{6} - \frac{13784}{507375} a^{5} + \frac{3299}{60885} a^{4} + \frac{508}{1845} a^{3} - \frac{162226}{507375} a^{2} + \frac{84257}{169125} a + \frac{1454}{5125}$, $\frac{1}{1356213375} a^{13} - \frac{8}{1356213375} a^{12} - \frac{179}{1356213375} a^{11} - \frac{223}{1356213375} a^{10} - \frac{5771}{271242675} a^{9} - \frac{4082491}{1356213375} a^{8} - \frac{621679}{452071125} a^{7} + \frac{1159391}{150690375} a^{6} - \frac{735191}{50230125} a^{5} - \frac{54488}{372075} a^{4} - \frac{4983931}{16743375} a^{3} + \frac{763127}{1860375} a^{2} - \frac{206633}{620125} a + \frac{23379}{56375}$, $\frac{1}{202268025455330109091460272781625} a^{14} + \frac{5713955004750860829982}{67422675151776703030486757593875} a^{13} - \frac{675311976582887673558308}{67422675151776703030486757593875} a^{12} - \frac{26471439061209433557124241}{202268025455330109091460272781625} a^{11} + \frac{18752314638721296987512632}{67422675151776703030486757593875} a^{10} - \frac{799792985745569711574074062}{67422675151776703030486757593875} a^{9} - \frac{337436584853852731232004495926}{202268025455330109091460272781625} a^{8} + \frac{19459337958202009812172290998}{6129334104706973002771523417625} a^{7} + \frac{36790281832100447362939922219}{22474225050592234343495585864625} a^{6} + \frac{1925943545598711012295624361}{277459568525830053623402294625} a^{5} - \frac{74581641162480322544864252321}{2497136116732470482610620651625} a^{4} - \frac{494449139755206768515951862356}{2497136116732470482610620651625} a^{3} - \frac{43638392871643878477391380709}{277459568525830053623402294625} a^{2} + \frac{81642539273627994293189811364}{277459568525830053623402294625} a - \frac{2786746186665322015887480478}{8407865712903941018890978625}$

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

Class group and class number

$C_{5}\times C_{15}$, which has order $75$ (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:  \( 190438426101124.5 \) (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

3.1.12300.2, 5.1.715270753125.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ R ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$5.15.17.3$x^{15} - 5 x^{13} + 10 x^{12} + 10 x^{11} + 10 x^{9} - 5 x^{8} - 10 x^{6} + 10 x^{5} + 5 x^{3} + 10$$15$$1$$17$$F_5 \times S_3$$[5/4]_{12}^{2}$
$41$41.5.4.1$x^{5} - 41$$5$$1$$4$$C_5$$[\ ]_{5}$
41.10.9.8$x^{10} + 318816$$10$$1$$9$$C_{10}$$[\ ]_{10}$