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Properties

Label 15.9.17517842124...0000.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{16}\cdot 5^{6}\cdot 17^{2}\cdot 37^{6}\cdot 2803^{4}\cdot 4003^{2}\cdot 152723^{2}$
Root discriminant $3040.18$
Ramified primes $2, 5, 17, 37, 2803, 4003, 152723$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T102

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22424000, 78484000, -145756000, -336640300, 624620520, -102417300, -284112164, 193325164, -54899712, 9994509, -1415004, 106045, -6552, 639, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 639*x^13 - 6552*x^12 + 106045*x^11 - 1415004*x^10 + 9994509*x^9 - 54899712*x^8 + 193325164*x^7 - 284112164*x^6 - 102417300*x^5 + 624620520*x^4 - 336640300*x^3 - 145756000*x^2 + 78484000*x + 22424000)
 
gp: K = bnfinit(x^15 + 639*x^13 - 6552*x^12 + 106045*x^11 - 1415004*x^10 + 9994509*x^9 - 54899712*x^8 + 193325164*x^7 - 284112164*x^6 - 102417300*x^5 + 624620520*x^4 - 336640300*x^3 - 145756000*x^2 + 78484000*x + 22424000, 1)
 

Normalized defining polynomial

\( x^{15} + 639 x^{13} - 6552 x^{12} + 106045 x^{11} - 1415004 x^{10} + 9994509 x^{9} - 54899712 x^{8} + 193325164 x^{7} - 284112164 x^{6} - 102417300 x^{5} + 624620520 x^{4} - 336640300 x^{3} - 145756000 x^{2} + 78484000 x + 22424000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-17517842124064884881429616734159236380876841984000000=-\,2^{16}\cdot 5^{6}\cdot 17^{2}\cdot 37^{6}\cdot 2803^{4}\cdot 4003^{2}\cdot 152723^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3040.18$
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:  $2, 5, 17, 37, 2803, 4003, 152723$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{40} a^{11} - \frac{1}{40} a^{9} - \frac{1}{20} a^{8} - \frac{3}{8} a^{7} + \frac{3}{20} a^{6} - \frac{11}{40} a^{5} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3} - \frac{7}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{40} a^{12} - \frac{1}{40} a^{10} - \frac{1}{20} a^{9} - \frac{3}{8} a^{8} + \frac{3}{20} a^{7} - \frac{11}{40} a^{6} - \frac{1}{20} a^{5} + \frac{1}{10} a^{4} - \frac{7}{20} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{200} a^{13} - \frac{1}{200} a^{11} - \frac{1}{100} a^{10} - \frac{3}{40} a^{9} - \frac{37}{100} a^{8} - \frac{91}{200} a^{7} - \frac{1}{100} a^{6} + \frac{1}{50} a^{5} + \frac{33}{100} a^{4} - \frac{3}{10} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{10915230617051366139959078082190756532346081060224806000} a^{14} + \frac{61066003555200136336285616721694876103454756848996}{136440382713142076749488476027384456654326013252810075} a^{13} - \frac{86711919195618863891690624826794258471526363397102491}{10915230617051366139959078082190756532346081060224806000} a^{12} - \frac{33046048763450653648150896850239934706198301329716291}{5457615308525683069979539041095378266173040530112403000} a^{11} + \frac{168804886957058477844795627028547624942511924838003}{2183046123410273227991815616438151306469216212044961200} a^{10} + \frac{108721087937261821395990145330285870290850193887828703}{5457615308525683069979539041095378266173040530112403000} a^{9} + \frac{3196844539739270154879584567623038885363605422729823639}{10915230617051366139959078082190756532346081060224806000} a^{8} + \frac{1650251010156294928041130085673147242993583885162332439}{5457615308525683069979539041095378266173040530112403000} a^{7} - \frac{169464108521917727691440800892836813452598705895539233}{5457615308525683069979539041095378266173040530112403000} a^{6} + \frac{1597257908881852256121290877593001270761984024793695433}{5457615308525683069979539041095378266173040530112403000} a^{5} - \frac{127960363766055471225219764927195049373435709574119761}{272880765426284153498976952054768913308652026505620150} a^{4} - \frac{112561427338818813891061597660562306825700235039157083}{545761530852568306997953904109537826617304053011240300} a^{3} - \frac{1000831890338649865565510180290884863824283360742651}{27288076542628415349897695205476891330865202650562015} a^{2} - \frac{5539566963984935314123176675584688123162486936283721}{54576153085256830699795390410953782661730405301124030} a + \frac{1885625875728472923334695053474074663302567080018582}{5457615308525683069979539041095378266173040530112403}$

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:  $11$
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:  \( 12427975138800000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T102:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10368000
The 140 conjugacy class representatives for [S(5)^3]S(3)=S(5)wrS(3) are not computed
Character table for [S(5)^3]S(3)=S(5)wrS(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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ R ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ R $15$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.10.7$x^{6} + 2 x^{5} + 4 x^{3} + 2$$6$$1$$10$$S_4\times C_2$$[2, 8/3, 8/3]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
37Data not computed
2803Data not computed
4003Data not computed
152723Data not computed