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Properties

Label 15.13.7717468344...4000.1
Degree $15$
Signature $[13, 1]$
Discriminant $-\,2^{15}\cdot 5^{3}\cdot 7^{10}\cdot 13^{4}\cdot 1973^{2}\cdot 9227^{4}\cdot 9097717^{2}$
Root discriminant $5319.02$
Ramified primes $2, 5, 7, 13, 1973, 9227, 9097717$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T101

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![119951000000000000, 53977950000000000, 6657280500000000, -422827275000000, -145380612000000, -4146108150000, 1121721226000, 69563456700, -4223348520, -388843381, 7903120, 1105179, -5920, -1628, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 1628*x^13 - 5920*x^12 + 1105179*x^11 + 7903120*x^10 - 388843381*x^9 - 4223348520*x^8 + 69563456700*x^7 + 1121721226000*x^6 - 4146108150000*x^5 - 145380612000000*x^4 - 422827275000000*x^3 + 6657280500000000*x^2 + 53977950000000000*x + 119951000000000000)
 
gp: K = bnfinit(x^15 - 1628*x^13 - 5920*x^12 + 1105179*x^11 + 7903120*x^10 - 388843381*x^9 - 4223348520*x^8 + 69563456700*x^7 + 1121721226000*x^6 - 4146108150000*x^5 - 145380612000000*x^4 - 422827275000000*x^3 + 6657280500000000*x^2 + 53977950000000000*x + 119951000000000000, 1)
 

Normalized defining polynomial

\( x^{15} - 1628 x^{13} - 5920 x^{12} + 1105179 x^{11} + 7903120 x^{10} - 388843381 x^{9} - 4223348520 x^{8} + 69563456700 x^{7} + 1121721226000 x^{6} - 4146108150000 x^{5} - 145380612000000 x^{4} - 422827275000000 x^{3} + 6657280500000000 x^{2} + 53977950000000000 x + 119951000000000000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[13, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-77174683442309589098663731731426228356468718082330624000=-\,2^{15}\cdot 5^{3}\cdot 7^{10}\cdot 13^{4}\cdot 1973^{2}\cdot 9227^{4}\cdot 9097717^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $5319.02$
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:  $2, 5, 7, 13, 1973, 9227, 9097717$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{10} a^{7} + \frac{1}{5} a^{5} - \frac{1}{10} a^{3} - \frac{1}{10} a$, $\frac{1}{100} a^{8} - \frac{7}{25} a^{6} - \frac{1}{5} a^{5} - \frac{21}{100} a^{4} + \frac{1}{5} a^{3} + \frac{19}{100} a^{2} - \frac{1}{5} a$, $\frac{1}{100000} a^{9} - \frac{7}{2500} a^{8} + \frac{43}{25000} a^{7} + \frac{212}{625} a^{6} - \frac{17621}{100000} a^{5} + \frac{9}{20} a^{4} + \frac{25219}{100000} a^{3} - \frac{623}{1250} a^{2} - \frac{27}{200} a + \frac{1}{10}$, $\frac{1}{25000000} a^{10} + \frac{11}{2500000} a^{9} - \frac{7257}{6250000} a^{8} - \frac{99}{25000} a^{7} + \frac{4771179}{25000000} a^{6} + \frac{477281}{2500000} a^{5} - \frac{2404781}{25000000} a^{4} + \frac{878557}{2500000} a^{3} + \frac{101689}{250000} a^{2} - \frac{951}{5000} a - \frac{31}{250}$, $\frac{1}{1000000000} a^{11} - \frac{1}{50000000} a^{10} + \frac{209}{125000000} a^{9} + \frac{13433}{12500000} a^{8} + \frac{25381179}{1000000000} a^{7} - \frac{9704023}{50000000} a^{6} + \frac{234184919}{1000000000} a^{5} + \frac{1614071}{10000000} a^{4} + \frac{264499}{5000000} a^{3} + \frac{46311}{125000} a^{2} + \frac{8151}{20000} a + \frac{353}{1000}$, $\frac{1}{50000000000} a^{12} + \frac{109}{6250000000} a^{10} - \frac{449}{625000000} a^{9} - \frac{233514821}{50000000000} a^{8} - \frac{22950711}{625000000} a^{7} - \frac{8067895881}{50000000000} a^{6} + \frac{332977037}{1250000000} a^{5} + \frac{41714771}{250000000} a^{4} - \frac{456661}{12500000} a^{3} - \frac{12661}{40000} a^{2} + \frac{2157}{6250} a + \frac{173}{500}$, $\frac{1}{2500000000000000} a^{13} - \frac{13}{2500000000000} a^{12} - \frac{26907}{625000000000000} a^{11} + \frac{843}{976562500000} a^{10} + \frac{3486633179}{2500000000000000} a^{9} - \frac{41607497597}{62500000000000} a^{8} + \frac{79957993622619}{2500000000000000} a^{7} + \frac{727179288191}{1953125000000} a^{6} - \frac{7408815433973}{25000000000000} a^{5} + \frac{488611893123}{1250000000000} a^{4} - \frac{15866029847}{50000000000} a^{3} + \frac{171534431}{1250000000} a^{2} - \frac{49947767}{100000000} a + \frac{1105867}{5000000}$, $\frac{1}{3200000000000000000} a^{14} - \frac{31}{160000000000000000} a^{13} + \frac{7238093}{800000000000000000} a^{12} + \frac{13059043}{40000000000000000} a^{11} - \frac{42616336421}{3200000000000000000} a^{10} - \frac{430679057393}{160000000000000000} a^{9} - \frac{13945272806411781}{3200000000000000000} a^{8} + \frac{116342434619077}{32000000000000000} a^{7} + \frac{7303964919564651}{32000000000000000} a^{6} - \frac{13152408738979}{100000000000000} a^{5} + \frac{60830350373313}{320000000000000} a^{4} - \frac{1482197361431}{3200000000000} a^{3} - \frac{216618916523}{640000000000} a^{2} + \frac{1248370297}{3200000000} a - \frac{132470827}{320000000}$

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:  $13$
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:  \( 139335852303000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T101:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 133 conjugacy class representatives for [S(5)^3]3=S(5)wr3 are not computed
Character table for [S(5)^3]3=S(5)wr3 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 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 R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $15$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $15$

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$$[\ ]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.9.8$x^{6} + 4 x^{2} - 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$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}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
13.5.0.1$x^{5} - 2 x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
1973Data not computed
9227Data not computed
9097717Data not computed