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Properties

Label 15.7.19674938756...3536.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{24}\cdot 3^{5}\cdot 7^{6}\cdot 107^{5}\cdot 193^{2}\cdot 8861^{2}$
Root discriminant $306.38$
Ramified primes $2, 3, 7, 107, 193, 8861$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T74

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53190273, -176039286, 61634071, 105081717, -17304477, -7122976, 1824339, -256343, -430836, -9442, -1270, -5254, -119, -26, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 26*x^13 - 119*x^12 - 5254*x^11 - 1270*x^10 - 9442*x^9 - 430836*x^8 - 256343*x^7 + 1824339*x^6 - 7122976*x^5 - 17304477*x^4 + 105081717*x^3 + 61634071*x^2 - 176039286*x + 53190273)
 
gp: K = bnfinit(x^15 - 3*x^14 - 26*x^13 - 119*x^12 - 5254*x^11 - 1270*x^10 - 9442*x^9 - 430836*x^8 - 256343*x^7 + 1824339*x^6 - 7122976*x^5 - 17304477*x^4 + 105081717*x^3 + 61634071*x^2 - 176039286*x + 53190273, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 26 x^{13} - 119 x^{12} - 5254 x^{11} - 1270 x^{10} - 9442 x^{9} - 430836 x^{8} - 256343 x^{7} + 1824339 x^{6} - 7122976 x^{5} - 17304477 x^{4} + 105081717 x^{3} + 61634071 x^{2} - 176039286 x + 53190273 \)

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:  \(19674938756461866304148089740812353536=2^{24}\cdot 3^{5}\cdot 7^{6}\cdot 107^{5}\cdot 193^{2}\cdot 8861^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $306.38$
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:  $2, 3, 7, 107, 193, 8861$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{6} - \frac{1}{2} a^{3} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{1381167986387300379810110883990147315862154731909463587372} a^{14} + \frac{36147034301350986406292590018339712266808964913214583858}{345291996596825094952527720997536828965538682977365896843} a^{13} - \frac{10719971570386143586581687433293585699912679149840143486}{345291996596825094952527720997536828965538682977365896843} a^{12} - \frac{136471309235850391690635766333320440350493307884491242579}{690583993193650189905055441995073657931077365954731793686} a^{11} - \frac{184582762660259456559173093214018755536756639953910810249}{1381167986387300379810110883990147315862154731909463587372} a^{10} - \frac{58307713413702685952199023971732679101224604143552845545}{345291996596825094952527720997536828965538682977365896843} a^{9} + \frac{28464355451754408650183659688946451339614881079114167233}{1381167986387300379810110883990147315862154731909463587372} a^{8} + \frac{344164995942158149001103707659304689272499093292845656823}{1381167986387300379810110883990147315862154731909463587372} a^{7} + \frac{317705605836488329164334130049522965052423395248990591001}{690583993193650189905055441995073657931077365954731793686} a^{6} + \frac{278549231600542763536651809864955287863292345467414310731}{690583993193650189905055441995073657931077365954731793686} a^{5} + \frac{30318369716106321688764090815864704629724363874444401082}{345291996596825094952527720997536828965538682977365896843} a^{4} - \frac{532377513758401649286746843112612258093317066912329240935}{1381167986387300379810110883990147315862154731909463587372} a^{3} + \frac{16569403422887746076065687592622539763377807858917561959}{62780363017604562718641403817733968902825215086793799426} a^{2} + \frac{542652430299112618025242391426716613285640490401896169375}{1381167986387300379810110883990147315862154731909463587372} a + \frac{13196884256183037808173024466819103756595548972816111470}{345291996596825094952527720997536828965538682977365896843}$

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:  \( 20870059861800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T74:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24000
The 40 conjugacy class representatives for [1/2.F(5)^3]S(3)
Character table for [1/2.F(5)^3]S(3) is not computed

Intermediate fields

3.3.321.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$$[\ ]^{3}$
2.12.24.225$x^{12} - 12 x^{11} + 16 x^{10} - 4 x^{9} - 10 x^{8} + 16 x^{7} - 8 x^{4} - 8 x^{2} + 8$$4$$3$$24$12T55$[2, 2, 3, 3]^{6}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.6.3$x^{8} - 7 x^{4} + 147$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
107Data not computed
193Data not computed
8861Data not computed