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Properties

Label 15.7.69454578208...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{15}\cdot 5^{6}\cdot 7^{10}\cdot 449^{4}\cdot 5503^{2}\cdot 19753^{2}$
Root discriminant $837.11$
Ramified primes $2, 5, 7, 449, 5503, 19753$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T95

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1839104000, 3218432000, -2988544000, -6132262400, 610352640, 2785791360, 777708416, -62206848, -36652608, -886392, 455488, 50420, -1296, -432, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 432*x^13 - 1296*x^12 + 50420*x^11 + 455488*x^10 - 886392*x^9 - 36652608*x^8 - 62206848*x^7 + 777708416*x^6 + 2785791360*x^5 + 610352640*x^4 - 6132262400*x^3 - 2988544000*x^2 + 3218432000*x + 1839104000)
 
gp: K = bnfinit(x^15 - 432*x^13 - 1296*x^12 + 50420*x^11 + 455488*x^10 - 886392*x^9 - 36652608*x^8 - 62206848*x^7 + 777708416*x^6 + 2785791360*x^5 + 610352640*x^4 - 6132262400*x^3 - 2988544000*x^2 + 3218432000*x + 1839104000, 1)
 

Normalized defining polynomial

\( x^{15} - 432 x^{13} - 1296 x^{12} + 50420 x^{11} + 455488 x^{10} - 886392 x^{9} - 36652608 x^{8} - 62206848 x^{7} + 777708416 x^{6} + 2785791360 x^{5} + 610352640 x^{4} - 6132262400 x^{3} - 2988544000 x^{2} + 3218432000 x + 1839104000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(69454578208162627938004535597516784128000000=2^{15}\cdot 5^{6}\cdot 7^{10}\cdot 449^{4}\cdot 5503^{2}\cdot 19753^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $837.11$
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:  $2, 5, 7, 449, 5503, 19753$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{16} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{8} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{9} + \frac{1}{16} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{256} a^{10} - \frac{1}{64} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{32} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{2560} a^{11} - \frac{3}{640} a^{9} + \frac{3}{320} a^{8} - \frac{3}{128} a^{7} + \frac{1}{20} a^{6} + \frac{11}{320} a^{5} - \frac{9}{80} a^{4} - \frac{9}{80} a^{3} - \frac{1}{40} a^{2}$, $\frac{1}{5120} a^{12} + \frac{1}{640} a^{10} + \frac{3}{640} a^{9} + \frac{1}{256} a^{8} - \frac{1}{160} a^{7} + \frac{21}{640} a^{6} + \frac{11}{160} a^{5} + \frac{3}{80} a^{4} - \frac{11}{80} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{51200} a^{13} + \frac{1}{6400} a^{11} - \frac{3}{1600} a^{10} + \frac{17}{2560} a^{9} + \frac{7}{800} a^{8} - \frac{99}{6400} a^{7} - \frac{21}{400} a^{6} + \frac{23}{800} a^{5} - \frac{9}{200} a^{4} - \frac{19}{80} a^{3} + \frac{3}{20} a^{2} - \frac{1}{4} a$, $\frac{1}{31890877281078994392370642187967145922095308800} a^{14} + \frac{30859611575548981505989013532155901414311}{3986359660134874299046330273495893240261913600} a^{13} - \frac{182435364685100716291592317559412797789849}{3986359660134874299046330273495893240261913600} a^{12} + \frac{108980051201719823865048328173946923960243}{1993179830067437149523165136747946620130956800} a^{11} + \frac{2098671476711081592932101009745446607512739}{1138959902895678371156094363855969497217689600} a^{10} + \frac{272438062844426249487027555287451070767219}{996589915033718574761582568373973310065478400} a^{9} + \frac{5863583687904927363025453885607374689114867}{569479951447839185578047181927984748608844800} a^{8} - \frac{5222727420867144183276366906643863883035203}{249147478758429643690395642093493327516369600} a^{7} + \frac{26733434075941206509707971415084975313876637}{498294957516859287380791284186986655032739200} a^{6} - \frac{1869936301611500316504163228454331835546627}{19165190673725357206953510930268717501259200} a^{5} + \frac{29944934978341837615710765571106596293256541}{249147478758429643690395642093493327516369600} a^{4} - \frac{598313067189413662871303592837321979784501}{3114343484480370546129945526168666593954620} a^{3} - \frac{57481386416852206263222213424113236325491}{958259533686267860347675546513435875062960} a^{2} - \frac{5172307720319187093724653619278770864899}{23956488342156696508691888662835896876574} a + \frac{72814832234186691564737101726683830411648}{155717174224018527306497276308433329697731}$

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:  $10$
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:  \( 33200333512000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T95:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1296000
The 53 conjugacy class representatives for [A(5)^3:2]3 are not computed
Character table for [A(5)^3:2]3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 sibling: 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 R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.8$x^{6} + 4 x^{2} - 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
449Data not computed
5503Data not computed
19753Data not computed