↑

Properties

Label 15.15.1970267782...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 3^{20}\cdot 5^{6}\cdot 17^{6}\cdot 1907^{4}\cdot 657388279^{2}$
Root discriminant $6601.38$
Ramified primes $2, 3, 5, 17, 1907, 657388279$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T98

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-132788224000, -697138176000, -1578520012800, -2011326630400, -1577524101120, -782507994240, -242693414016, -44677501536, -4227619392, -78499800, 18581472, 1129752, -16848, -2106, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2106*x^13 - 16848*x^12 + 1129752*x^11 + 18581472*x^10 - 78499800*x^9 - 4227619392*x^8 - 44677501536*x^7 - 242693414016*x^6 - 782507994240*x^5 - 1577524101120*x^4 - 2011326630400*x^3 - 1578520012800*x^2 - 697138176000*x - 132788224000)
 
gp: K = bnfinit(x^15 - 2106*x^13 - 16848*x^12 + 1129752*x^11 + 18581472*x^10 - 78499800*x^9 - 4227619392*x^8 - 44677501536*x^7 - 242693414016*x^6 - 782507994240*x^5 - 1577524101120*x^4 - 2011326630400*x^3 - 1578520012800*x^2 - 697138176000*x - 132788224000, 1)
 

Normalized defining polynomial

\( x^{15} - 2106 x^{13} - 16848 x^{12} + 1129752 x^{11} + 18581472 x^{10} - 78499800 x^{9} - 4227619392 x^{8} - 44677501536 x^{7} - 242693414016 x^{6} - 782507994240 x^{5} - 1577524101120 x^{4} - 2011326630400 x^{3} - 1578520012800 x^{2} - 697138176000 x - 132788224000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1970267782154216617690093863120316160444627676893184000000=2^{18}\cdot 3^{20}\cdot 5^{6}\cdot 17^{6}\cdot 1907^{4}\cdot 657388279^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $6601.38$
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:  $2, 3, 5, 17, 1907, 657388279$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{6} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{80} a^{7} - \frac{1}{40} a^{5} + \frac{1}{20} a^{4} - \frac{1}{2} a$, $\frac{1}{800} a^{8} + \frac{3}{400} a^{6} + \frac{1}{100} a^{5} + \frac{2}{25} a^{4} - \frac{1}{50} a^{3} + \frac{7}{100} a^{2} - \frac{1}{5}$, $\frac{1}{3200} a^{9} + \frac{3}{1600} a^{7} - \frac{1}{100} a^{6} - \frac{7}{400} a^{5} + \frac{1}{50} a^{4} + \frac{17}{400} a^{3} + \frac{1}{10} a^{2} + \frac{9}{20} a$, $\frac{1}{108800} a^{10} - \frac{1}{2176} a^{8} - \frac{21}{3400} a^{7} - \frac{169}{13600} a^{6} - \frac{3}{850} a^{5} + \frac{17}{800} a^{4} - \frac{3}{100} a^{3} + \frac{11}{200} a^{2} + \frac{1}{5}$, $\frac{1}{2176000} a^{11} + \frac{77}{1088000} a^{9} + \frac{77}{136000} a^{8} - \frac{89}{68000} a^{7} - \frac{709}{68000} a^{6} + \frac{49}{3200} a^{5} + \frac{161}{2000} a^{4} + \frac{163}{2000} a^{3} - \frac{181}{1000} a^{2} + \frac{29}{200} a + \frac{13}{50}$, $\frac{1}{21760000} a^{12} - \frac{43}{10880000} a^{10} - \frac{89}{680000} a^{9} - \frac{7}{21250} a^{8} - \frac{767}{340000} a^{7} - \frac{6559}{544000} a^{6} + \frac{5707}{340000} a^{5} - \frac{947}{20000} a^{4} - \frac{89}{2500} a^{3} - \frac{427}{2000} a^{2} - \frac{111}{250} a + \frac{7}{25}$, $\frac{1}{87040000000} a^{13} - \frac{31}{2176000000} a^{12} + \frac{4827}{43520000000} a^{11} + \frac{1681}{2720000000} a^{10} - \frac{1003961}{10880000000} a^{9} - \frac{1028459}{2720000000} a^{8} - \frac{10360719}{2176000000} a^{7} - \frac{2094881}{170000000} a^{6} - \frac{9584183}{2720000000} a^{5} + \frac{297329}{40000000} a^{4} + \frac{1367743}{8000000} a^{3} + \frac{93673}{1000000} a^{2} - \frac{53993}{400000} a + \frac{13241}{100000}$, $\frac{1}{11141120000000} a^{14} - \frac{11}{2785280000000} a^{13} + \frac{59307}{5570560000000} a^{12} - \frac{250003}{1392640000000} a^{11} + \frac{277879}{81920000000} a^{10} + \frac{25474601}{174080000000} a^{9} - \frac{338383451}{1392640000000} a^{8} - \frac{1401121001}{348160000000} a^{7} + \frac{3782363001}{348160000000} a^{6} - \frac{29798443}{1280000000} a^{5} - \frac{501167801}{5120000000} a^{4} + \frac{21874903}{256000000} a^{3} + \frac{29569851}{256000000} a^{2} - \frac{3010933}{6400000} a + \frac{1305059}{3200000}$

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:  $14$
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:  \( 955284233958000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T98:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 71 conjugacy class representatives for [1/2.S(5)^3]3 are not computed
Character table for [1/2.S(5)^3]3 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\)

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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ $15$ R ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{7}$ $15$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $15$

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$$[\ ]^{3}$
2.6.9.6$x^{6} + 4 x^{2} + 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.9.6$x^{6} + 4 x^{2} + 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
3Data not computed
$5$5.3.0.1$x^{3} - x + 2$$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}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.5.4.1$x^{5} - 17$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
1907Data not computed
657388279Data not computed