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Properties

Label 15.3.45924806577...6161.1
Degree $15$
Signature $[3, 6]$
Discriminant $3^{20}\cdot 19^{10}\cdot 59^{2}\cdot 61^{6}\cdot 3461^{2}$
Root discriminant $814.34$
Ramified primes $3, 19, 59, 61, 3461$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T67

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13824, 34560, 36864, -151552, -210144, 645456, -465624, -22668, 130602, -21029, -11358, 2640, -76, 27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 27*x^13 - 76*x^12 + 2640*x^11 - 11358*x^10 - 21029*x^9 + 130602*x^8 - 22668*x^7 - 465624*x^6 + 645456*x^5 - 210144*x^4 - 151552*x^3 + 36864*x^2 + 34560*x - 13824)
 
gp: K = bnfinit(x^15 + 27*x^13 - 76*x^12 + 2640*x^11 - 11358*x^10 - 21029*x^9 + 130602*x^8 - 22668*x^7 - 465624*x^6 + 645456*x^5 - 210144*x^4 - 151552*x^3 + 36864*x^2 + 34560*x - 13824, 1)
 

Normalized defining polynomial

\( x^{15} + 27 x^{13} - 76 x^{12} + 2640 x^{11} - 11358 x^{10} - 21029 x^{9} + 130602 x^{8} - 22668 x^{7} - 465624 x^{6} + 645456 x^{5} - 210144 x^{4} - 151552 x^{3} + 36864 x^{2} + 34560 x - 13824 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(45924806577969328401107322831381126116656161=3^{20}\cdot 19^{10}\cdot 59^{2}\cdot 61^{6}\cdot 3461^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $814.34$
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:  $3, 19, 59, 61, 3461$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{7} + \frac{1}{10} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{3}{10} a^{3} - \frac{1}{2} a + \frac{1}{5}$, $\frac{1}{20} a^{8} + \frac{1}{20} a^{6} - \frac{1}{10} a^{3} + \frac{1}{4} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{40} a^{9} - \frac{1}{40} a^{7} + \frac{1}{5} a^{6} - \frac{1}{10} a^{5} - \frac{3}{20} a^{4} + \frac{11}{40} a^{3} - \frac{9}{20} a^{2} - \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{480} a^{10} + \frac{1}{160} a^{8} + \frac{1}{40} a^{7} + \frac{1}{20} a^{6} + \frac{7}{80} a^{5} - \frac{101}{480} a^{4} + \frac{7}{80} a^{3} - \frac{11}{40} a^{2} - \frac{11}{30} a - \frac{1}{2}$, $\frac{1}{960} a^{11} + \frac{1}{320} a^{9} + \frac{1}{80} a^{8} + \frac{1}{40} a^{7} + \frac{7}{160} a^{6} + \frac{139}{960} a^{5} - \frac{33}{160} a^{4} + \frac{29}{80} a^{3} + \frac{1}{15} a^{2} - \frac{1}{4} a$, $\frac{1}{1920} a^{12} - \frac{1}{1920} a^{10} + \frac{1}{160} a^{9} + \frac{1}{160} a^{8} - \frac{1}{320} a^{7} - \frac{437}{1920} a^{6} + \frac{19}{320} a^{5} - \frac{13}{120} a^{4} - \frac{73}{240} a^{3} - \frac{7}{20} a^{2} - \frac{2}{15} a - \frac{1}{2}$, $\frac{1}{115200} a^{13} - \frac{1}{4800} a^{12} + \frac{13}{38400} a^{11} + \frac{11}{28800} a^{10} - \frac{61}{9600} a^{9} - \frac{127}{6400} a^{8} - \frac{1157}{115200} a^{7} + \frac{2239}{19200} a^{6} - \frac{869}{4800} a^{5} + \frac{179}{1200} a^{4} + \frac{101}{800} a^{3} - \frac{97}{400} a^{2} + \frac{389}{1800} a + \frac{11}{300}$, $\frac{1}{10723006582272000} a^{14} - \frac{1358685697}{1787167763712000} a^{13} - \frac{154646143961}{1191445175808000} a^{12} + \frac{1196070478621}{5361503291136000} a^{11} + \frac{669053119}{15676910208000} a^{10} - \frac{281426343767}{119144517580800} a^{9} + \frac{159768094420831}{10723006582272000} a^{8} - \frac{2069399498531}{44679194092800} a^{7} + \frac{1189751909551}{893583881856000} a^{6} + \frac{2496764821781}{24821774496000} a^{5} + \frac{39095392850569}{223395970464000} a^{4} - \frac{2432897193323}{6981124077000} a^{3} + \frac{504161096437}{20943372231000} a^{2} + \frac{627537873199}{1745281019250} a + \frac{1040221793567}{4654082718000}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  $8$
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:  \( 14382385575100000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T67:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12000
The 32 conjugacy class representatives for [1/2.F(5)^3]3
Character table for [1/2.F(5)^3]3 is not computed

Intermediate fields

3.3.29241.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 siblings: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{5}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $15$ R

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
$3$3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.6.8.6$x^{6} + 18 x^{2} + 36$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.6$x^{6} + 18 x^{2} + 36$$3$$2$$8$$C_6$$[2]^{2}$
$19$19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.2$x^{4} - 59 x^{2} + 6962$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
59.4.0.1$x^{4} - x + 14$$1$$4$$0$$C_4$$[\ ]^{4}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$
3461Data not computed