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Properties

Label 15.1.61848148034...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{12}\cdot 5^{15}\cdot 11^{12}\cdot 6911^{5}$
Root discriminant $1129.16$
Ramified primes $2, 5, 11, 6911$
Class number $125$ (GRH)
Class group $[5, 5, 5]$ (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![-35925416, -48405120, -5059000, 5331200, -11509040, 11711, -328800, -284155, -2280, 10230, 376, 10, -160, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^13 - 160*x^12 + 10*x^11 + 376*x^10 + 10230*x^9 - 2280*x^8 - 284155*x^7 - 328800*x^6 + 11711*x^5 - 11509040*x^4 + 5331200*x^3 - 5059000*x^2 - 48405120*x - 35925416)
 
gp: K = bnfinit(x^15 - 5*x^13 - 160*x^12 + 10*x^11 + 376*x^10 + 10230*x^9 - 2280*x^8 - 284155*x^7 - 328800*x^6 + 11711*x^5 - 11509040*x^4 + 5331200*x^3 - 5059000*x^2 - 48405120*x - 35925416, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{13} - 160 x^{12} + 10 x^{11} + 376 x^{10} + 10230 x^{9} - 2280 x^{8} - 284155 x^{7} - 328800 x^{6} + 11711 x^{5} - 11509040 x^{4} + 5331200 x^{3} - 5059000 x^{2} - 48405120 x - 35925416 \)

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:  \(-6184814803411251939738171243533875000000000000=-\,2^{12}\cdot 5^{15}\cdot 11^{12}\cdot 6911^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1129.16$
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:  $2, 5, 11, 6911$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{9} + \frac{3}{16} a^{7} + \frac{3}{16} a^{5} - \frac{1}{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{16} a^{8} - \frac{1}{4} a^{7} + \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{13} + \frac{1}{16} a^{9} - \frac{1}{4} a^{6} - \frac{3}{32} a^{5} + \frac{1}{4}$, $\frac{1}{1645772263067968472574580442813952418719392} a^{14} + \frac{25322468167267683676714913158852415551893}{1645772263067968472574580442813952418719392} a^{13} - \frac{598967109835204225630169628723988560727}{822886131533984236287290221406976209359696} a^{12} - \frac{2622644152222439526705857651005448614633}{205721532883496059071822555351744052339924} a^{11} - \frac{17307693548178403119057214665080893074525}{411443065766992118143645110703488104679848} a^{10} + \frac{53932523709715097951523328156699166006765}{822886131533984236287290221406976209359696} a^{9} + \frac{71734705890913705013098709230345387813219}{822886131533984236287290221406976209359696} a^{8} + \frac{22633731043723437179029518239020885054315}{102860766441748029535911277675872026169962} a^{7} - \frac{191716376827485144469661892850011024934509}{1645772263067968472574580442813952418719392} a^{6} + \frac{119170790213495995102358702760041171424593}{1645772263067968472574580442813952418719392} a^{5} + \frac{3834502579058810471083856912092671715722}{51430383220874014767955638837936013084981} a^{4} - \frac{15378066737913365109175982525301616757345}{102860766441748029535911277675872026169962} a^{3} + \frac{23870401415111616043651147320732322336181}{102860766441748029535911277675872026169962} a^{2} - \frac{23161157510805109904870872517328344770325}{205721532883496059071822555351744052339924} a - \frac{44151453244279578227648527872431514055407}{205721532883496059071822555351744052339924}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{5}$, which has order $125$ (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:  \( 3869041310693481.0 \) (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

3.1.6911.1, 5.1.732050000.1

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 ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$5$5.15.15.40$x^{15} + 10 x^{14} + 5 x^{13} + 20 x^{12} + 20 x^{11} + 17 x^{10} + 10 x^{9} + 15 x^{8} + 12 x^{5} + 15 x^{3} + 10 x^{2} + 20 x + 17$$5$$3$$15$$F_5\times C_3$$[5/4]_{4}^{3}$
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
6911Data not computed