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Properties

Label 15.11.4653544347...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{9}\cdot 19^{2}\cdot 61^{2}\cdot 67^{5}\cdot 97^{4}\cdot 132421^{2}$
Root discriminant $2387.00$
Ramified primes $2, 5, 7, 19, 61, 67, 97, 132421$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T100

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2781184000, -14601216000, -33061324800, -42126246400, -33040465920, -16395827840, -5106134656, -967758176, -110763072, -9568440, -943808, -59528, -288, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 36*x^13 - 288*x^12 - 59528*x^11 - 943808*x^10 - 9568440*x^9 - 110763072*x^8 - 967758176*x^7 - 5106134656*x^6 - 16395827840*x^5 - 33040465920*x^4 - 42126246400*x^3 - 33061324800*x^2 - 14601216000*x - 2781184000)
 
gp: K = bnfinit(x^15 - 36*x^13 - 288*x^12 - 59528*x^11 - 943808*x^10 - 9568440*x^9 - 110763072*x^8 - 967758176*x^7 - 5106134656*x^6 - 16395827840*x^5 - 33040465920*x^4 - 42126246400*x^3 - 33061324800*x^2 - 14601216000*x - 2781184000, 1)
 

Normalized defining polynomial

\( x^{15} - 36 x^{13} - 288 x^{12} - 59528 x^{11} - 943808 x^{10} - 9568440 x^{9} - 110763072 x^{8} - 967758176 x^{7} - 5106134656 x^{6} - 16395827840 x^{5} - 33040465920 x^{4} - 42126246400 x^{3} - 33061324800 x^{2} - 14601216000 x - 2781184000 \)

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:  \(465354434733444657917549052039955476309397504000000=2^{18}\cdot 5^{6}\cdot 7^{9}\cdot 19^{2}\cdot 61^{2}\cdot 67^{5}\cdot 97^{4}\cdot 132421^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2387.00$
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:  $2, 5, 7, 19, 61, 67, 97, 132421$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{6} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{80} a^{7} + \frac{1}{20} a^{4} + \frac{1}{5} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{800} a^{8} - \frac{1}{200} a^{6} + \frac{1}{100} a^{5} - \frac{1}{50} a^{4} + \frac{9}{50} a^{3} + \frac{7}{100} a^{2} - \frac{1}{5}$, $\frac{1}{3200} a^{9} - \frac{1}{800} a^{7} - \frac{1}{100} a^{6} - \frac{7}{400} a^{5} + \frac{7}{100} a^{4} + \frac{97}{400} a^{3} + \frac{1}{5} a^{2} + \frac{9}{20} a$, $\frac{1}{6400} a^{10} - \frac{1}{1600} a^{8} - \frac{1}{200} a^{7} - \frac{7}{800} a^{6} + \frac{1}{100} a^{5} + \frac{97}{800} a^{4} + \frac{3}{20} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{128000} a^{11} - \frac{1}{8000} a^{9} + \frac{1}{4000} a^{8} + \frac{79}{16000} a^{7} - \frac{7}{2000} a^{6} - \frac{79}{3200} a^{5} - \frac{133}{2000} a^{4} + \frac{191}{2000} a^{3} - \frac{227}{1000} a^{2} + \frac{73}{200} a + \frac{21}{50}$, $\frac{1}{2560000} a^{12} - \frac{1}{256000} a^{11} - \frac{1}{160000} a^{10} - \frac{1}{20000} a^{9} + \frac{79}{320000} a^{8} + \frac{657}{160000} a^{7} - \frac{703}{64000} a^{6} - \frac{1277}{160000} a^{5} + \frac{3521}{40000} a^{4} + \frac{519}{10000} a^{3} + \frac{47}{800} a^{2} + \frac{917}{2000} a + \frac{37}{100}$, $\frac{1}{5120000000} a^{13} + \frac{19}{128000000} a^{12} - \frac{2069}{1280000000} a^{11} - \frac{2007}{80000000} a^{10} + \frac{91479}{640000000} a^{9} - \frac{21681}{40000000} a^{8} - \frac{3991}{1024000} a^{7} + \frac{336391}{40000000} a^{6} + \frac{3248697}{160000000} a^{5} + \frac{1719813}{40000000} a^{4} + \frac{945571}{8000000} a^{3} - \frac{237669}{1000000} a^{2} + \frac{191779}{400000} a + \frac{20477}{100000}$, $\frac{1}{655360000000} a^{14} - \frac{11}{163840000000} a^{13} + \frac{30171}{163840000000} a^{12} - \frac{129659}{40960000000} a^{11} - \frac{2679497}{81920000000} a^{10} - \frac{634003}{20480000000} a^{9} + \frac{30070009}{81920000000} a^{8} - \frac{41095061}{20480000000} a^{7} - \frac{74859759}{20480000000} a^{6} - \frac{6720723}{640000000} a^{5} + \frac{143868203}{5120000000} a^{4} + \frac{10720891}{256000000} a^{3} + \frac{43085647}{256000000} a^{2} - \frac{2288801}{6400000} a - \frac{478377}{3200000}$

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:  \( 2356986138750000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T100:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 79 conjugacy class representatives for [1/2.S(5)^3]S(3) are not computed
Character table for [1/2.S(5)^3]S(3) is not computed

Intermediate fields

3.3.469.1

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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.6.6$x^{4} - 20$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
2.4.6.5$x^{4} + 2 x^{2} - 4$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
$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}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.5.0.1$x^{5} - x + 4$$1$$5$$0$$C_5$$[\ ]^{5}$
7.10.9.1$x^{10} - 7$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.10.0.1$x^{10} + x^{2} - 2 x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
61Data not computed
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.3.0.1$x^{3} - x + 16$$1$$3$$0$$C_3$$[\ ]^{3}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.5.4.1$x^{5} - 97$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
132421Data not computed