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Properties

Label 15.11.4041163971...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{21}\cdot 5^{6}\cdot 31^{4}\cdot 37^{5}\cdot 1387715921^{2}$
Root discriminant $692.52$
Ramified primes $2, 5, 31, 37, 1387715921$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T97

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3968000, -41664000, -168640000, -324408800, -285963840, -73028000, 27161712, 10678864, -1459296, -370134, 75392, 5540, -936, -144, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 144*x^13 - 936*x^12 + 5540*x^11 + 75392*x^10 - 370134*x^9 - 1459296*x^8 + 10678864*x^7 + 27161712*x^6 - 73028000*x^5 - 285963840*x^4 - 324408800*x^3 - 168640000*x^2 - 41664000*x - 3968000)
 
gp: K = bnfinit(x^15 - 144*x^13 - 936*x^12 + 5540*x^11 + 75392*x^10 - 370134*x^9 - 1459296*x^8 + 10678864*x^7 + 27161712*x^6 - 73028000*x^5 - 285963840*x^4 - 324408800*x^3 - 168640000*x^2 - 41664000*x - 3968000, 1)
 

Normalized defining polynomial

\( x^{15} - 144 x^{13} - 936 x^{12} + 5540 x^{11} + 75392 x^{10} - 370134 x^{9} - 1459296 x^{8} + 10678864 x^{7} + 27161712 x^{6} - 73028000 x^{5} - 285963840 x^{4} - 324408800 x^{3} - 168640000 x^{2} - 41664000 x - 3968000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4041163971392929828955494668759105536000000=2^{21}\cdot 5^{6}\cdot 31^{4}\cdot 37^{5}\cdot 1387715921^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $692.52$
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:  $2, 5, 31, 37, 1387715921$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{16} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{3}{8} a^{4} - \frac{1}{2} a$, $\frac{1}{160} a^{11} - \frac{1}{40} a^{9} + \frac{1}{40} a^{8} + \frac{1}{8} a^{7} + \frac{1}{5} a^{6} - \frac{27}{80} a^{5} - \frac{1}{10} a^{4} - \frac{7}{20} a^{3} + \frac{9}{20} a^{2}$, $\frac{1}{320} a^{12} - \frac{1}{80} a^{10} + \frac{1}{80} a^{9} + \frac{1}{16} a^{8} + \frac{1}{10} a^{7} - \frac{27}{160} a^{6} + \frac{9}{20} a^{5} - \frac{7}{40} a^{4} - \frac{11}{40} a^{3}$, $\frac{1}{3200} a^{13} - \frac{1}{800} a^{11} - \frac{9}{800} a^{10} + \frac{9}{160} a^{9} + \frac{11}{100} a^{8} - \frac{267}{1600} a^{7} - \frac{1}{200} a^{6} - \frac{87}{400} a^{5} + \frac{59}{400} a^{4} - \frac{1}{5} a^{3} - \frac{3}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{4131812288005219384753008298978190322784000} a^{14} + \frac{350237544233537795099435276864366577}{51647653600065242309412603737227379034800} a^{13} - \frac{92140152807231542560471165615077372591}{1032953072001304846188252074744547580696000} a^{12} + \frac{692527248922468162006652741089362508339}{258238268000326211547063018686136895174000} a^{11} - \frac{148145994294404946390454897618072553907}{206590614400260969237650414948909516139200} a^{10} - \frac{10400076086184434610720369162007484696493}{258238268000326211547063018686136895174000} a^{9} + \frac{64220913079469509786379696387055658530893}{2065906144002609692376504149489095161392000} a^{8} + \frac{14864516660567041274889075101520887122707}{129119134000163105773531509343068447587000} a^{7} - \frac{19623570838400368106780087411017509562277}{516476536000652423094126037372273790348000} a^{6} - \frac{43846573854025865352682482354963746584679}{129119134000163105773531509343068447587000} a^{5} + \frac{12001723028963414127433003365207528723599}{25823826800032621154706301868613689517400} a^{4} + \frac{11804374033306179086649525771857783465617}{25823826800032621154706301868613689517400} a^{3} + \frac{1506962998008160308714079973667539063319}{5164765360006524230941260373722737903480} a^{2} - \frac{207684162740664489977252831284819422687}{1291191340001631057735315093430684475870} a - \frac{29006976302296613471039474927194284951}{129119134000163105773531509343068447587}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $12$
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:  \( 182131728633000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T97:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 70 conjugacy class representatives for [A(5)^3:2]S(3) are not computed
Character table for [A(5)^3:2]S(3) is not computed

Intermediate fields

3.3.148.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ R R $15$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$[\ ]_{3}^{2}$
2.6.11.14$x^{6} + 2 x^{4} + 10$$6$$1$$11$$S_4\times C_2$$[4/3, 4/3, 3]_{3}^{2}$
2.6.8.4$x^{6} + 2 x^{3} + 2 x^{2} + 2$$6$$1$$8$$S_4\times C_2$$[4/3, 4/3, 2]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.5.4.3$x^{5} - 1519$$5$$1$$4$$C_5$$[\ ]_{5}$
37Data not computed
1387715921Data not computed