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Properties

Label 15.3.14068573387...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{23}\cdot 5^{17}\cdot 89^{13}$
Root discriminant $877.44$
Ramified primes $2, 5, 89$
Class number $160$ (GRH)
Class group $[4, 40]$ (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![-520184176, 420777040, 231378160, -5194480, 68720, -2453272, 1638320, -976120, 392310, -24090, -15553, 2265, 460, -80, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 80*x^13 + 460*x^12 + 2265*x^11 - 15553*x^10 - 24090*x^9 + 392310*x^8 - 976120*x^7 + 1638320*x^6 - 2453272*x^5 + 68720*x^4 - 5194480*x^3 + 231378160*x^2 + 420777040*x - 520184176)
 
gp: K = bnfinit(x^15 - 5*x^14 - 80*x^13 + 460*x^12 + 2265*x^11 - 15553*x^10 - 24090*x^9 + 392310*x^8 - 976120*x^7 + 1638320*x^6 - 2453272*x^5 + 68720*x^4 - 5194480*x^3 + 231378160*x^2 + 420777040*x - 520184176, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 80 x^{13} + 460 x^{12} + 2265 x^{11} - 15553 x^{10} - 24090 x^{9} + 392310 x^{8} - 976120 x^{7} + 1638320 x^{6} - 2453272 x^{5} + 68720 x^{4} - 5194480 x^{3} + 231378160 x^{2} + 420777040 x - 520184176 \)

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:  \(140685733870773315116885561600000000000000000=2^{23}\cdot 5^{17}\cdot 89^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $877.44$
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:  $2, 5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{10} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{7} - \frac{1}{2} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{10} a^{8} - \frac{1}{2} a^{4} + \frac{2}{5} a^{3}$, $\frac{1}{50} a^{9} + \frac{1}{50} a^{8} + \frac{1}{50} a^{7} + \frac{1}{50} a^{6} - \frac{2}{25} a^{5} + \frac{2}{25} a^{4} + \frac{2}{25} a^{3} + \frac{2}{25} a^{2} + \frac{2}{25} a - \frac{8}{25}$, $\frac{1}{8900} a^{10} - \frac{13}{8900} a^{9} + \frac{53}{2225} a^{8} + \frac{63}{2225} a^{7} + \frac{287}{8900} a^{6} - \frac{107}{1780} a^{5} - \frac{888}{2225} a^{4} - \frac{588}{2225} a^{3} - \frac{898}{2225} a^{2} - \frac{738}{2225} a + \frac{161}{2225}$, $\frac{1}{8900} a^{11} + \frac{43}{8900} a^{9} + \frac{169}{4450} a^{8} + \frac{3}{8900} a^{7} - \frac{91}{2225} a^{6} + \frac{173}{8900} a^{5} + \frac{211}{4450} a^{4} - \frac{87}{2225} a^{3} - \frac{397}{2225} a^{2} + \frac{357}{2225} a - \frac{577}{2225}$, $\frac{1}{44500} a^{12} - \frac{1}{22250} a^{11} + \frac{1}{44500} a^{10} - \frac{27}{4450} a^{9} - \frac{349}{8900} a^{8} + \frac{219}{22250} a^{7} + \frac{239}{44500} a^{6} - \frac{831}{22250} a^{5} - \frac{379}{2225} a^{4} + \frac{676}{2225} a^{3} + \frac{1309}{11125} a^{2} + \frac{5497}{11125} a + \frac{1334}{11125}$, $\frac{1}{89000} a^{13} - \frac{1}{89000} a^{12} + \frac{1}{22250} a^{11} - \frac{1}{22250} a^{10} - \frac{159}{17800} a^{9} + \frac{3163}{89000} a^{8} + \frac{361}{44500} a^{7} - \frac{1419}{44500} a^{6} - \frac{969}{11125} a^{5} - \frac{159}{4450} a^{4} + \frac{6409}{22250} a^{3} + \frac{238}{11125} a^{2} + \frac{2198}{11125} a + \frac{2757}{11125}$, $\frac{1}{9044983275352457808447509724761000} a^{14} + \frac{49106609225112113527287397041}{9044983275352457808447509724761000} a^{13} + \frac{9102707840941707628335864249}{1130622909419057226055938715595125} a^{12} + \frac{4039003502677186448376505837}{4522491637676228904223754862380500} a^{11} - \frac{5136224739418851798191390447}{101629025565757952903904603649000} a^{10} - \frac{67563679538156252634320077499427}{9044983275352457808447509724761000} a^{9} - \frac{57917211743901688732828250312291}{4522491637676228904223754862380500} a^{8} - \frac{36711114667247162831194409890103}{1130622909419057226055938715595125} a^{7} - \frac{42101677296412792165231479549721}{1130622909419057226055938715595125} a^{6} + \frac{87096457171757044142403136130919}{2261245818838114452111877431190250} a^{5} + \frac{615913780977266484120252803213319}{2261245818838114452111877431190250} a^{4} + \frac{493766660320779266886310525779877}{1130622909419057226055938715595125} a^{3} + \frac{80806538674511241766076941809604}{1130622909419057226055938715595125} a^{2} + \frac{174675251659561390005435757717893}{1130622909419057226055938715595125} a + \frac{543491202631732019159366184455404}{1130622909419057226055938715595125}$

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

Class group and class number

$C_{4}\times C_{40}$, which has order $160$ (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:  \( 308642886210973.75 \) (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

3.3.17800.1, 5.1.3137112050000.2

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]_{5}^{4}$
2.10.19.1$x^{10} - 2$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{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$$[5/4]_{12}^{2}$
$89$89.5.4.1$x^{5} - 89$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
89.10.9.1$x^{10} - 89$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$