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Properties

Label 15.1.28281005788...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 5^{10}\cdot 11^{13}$
Root discriminant $67.62$
Ramified primes $2, 5, 11$
Class number $5$ (GRH)
Class group $[5]$ (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![176, 704, 176, 2728, -2112, 4940, -4832, 5212, -3184, 2556, -672, 278, -108, 16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 16*x^13 - 108*x^12 + 278*x^11 - 672*x^10 + 2556*x^9 - 3184*x^8 + 5212*x^7 - 4832*x^6 + 4940*x^5 - 2112*x^4 + 2728*x^3 + 176*x^2 + 704*x + 176)
 
gp: K = bnfinit(x^15 - 4*x^14 + 16*x^13 - 108*x^12 + 278*x^11 - 672*x^10 + 2556*x^9 - 3184*x^8 + 5212*x^7 - 4832*x^6 + 4940*x^5 - 2112*x^4 + 2728*x^3 + 176*x^2 + 704*x + 176, 1)
 

Normalized defining polynomial

\( x^{15} - 4 x^{14} + 16 x^{13} - 108 x^{12} + 278 x^{11} - 672 x^{10} + 2556 x^{9} - 3184 x^{8} + 5212 x^{7} - 4832 x^{6} + 4940 x^{5} - 2112 x^{4} + 2728 x^{3} + 176 x^{2} + 704 x + 176 \)

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:  \(-2828100578830827520000000000=-\,2^{23}\cdot 5^{10}\cdot 11^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.62$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{10} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{10} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{20} a^{10} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{20} a^{11} - \frac{1}{10} a^{8} + \frac{1}{10} a^{7} + \frac{2}{5} a^{4} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{20} a^{12} + \frac{1}{10} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{3160} a^{13} - \frac{3}{158} a^{12} - \frac{6}{395} a^{11} - \frac{33}{1580} a^{10} - \frac{51}{1580} a^{9} + \frac{7}{790} a^{8} + \frac{53}{395} a^{7} - \frac{127}{790} a^{6} + \frac{23}{790} a^{5} + \frac{38}{79} a^{4} + \frac{111}{790} a^{3} + \frac{188}{395} a^{2} - \frac{112}{395} a - \frac{46}{395}$, $\frac{1}{458285744846200} a^{14} + \frac{9697266837}{458285744846200} a^{13} - \frac{413194554183}{114571436211550} a^{12} + \frac{353080033727}{22914287242310} a^{11} - \frac{247143652333}{114571436211550} a^{10} + \frac{10789193644233}{229142872423100} a^{9} - \frac{904702134997}{114571436211550} a^{8} + \frac{11163558697346}{57285718105775} a^{7} - \frac{2178388641897}{22914287242310} a^{6} + \frac{1534195960891}{57285718105775} a^{5} + \frac{234905443793}{640063889450} a^{4} - \frac{42004080793951}{114571436211550} a^{3} - \frac{16370941931332}{57285718105775} a^{2} + \frac{2183720705579}{11457143621155} a + \frac{17979713560408}{57285718105775}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 45886304.85748534 \) (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

3.1.440.1, 5.1.29282000.1

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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.19.1$x^{10} - 2$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{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}$
$11$11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.2$x^{10} - 72171$$10$$1$$9$$C_{10}$$[\ ]_{10}$