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Properties

Label 15.11.2884306892...5696.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{20}\cdot 37^{5}\cdot 43^{2}\cdot 139^{2}\cdot 317^{3}\cdot 1867^{2}$
Root discriminant $231.21$
Ramified primes $2, 37, 43, 139, 317, 1867$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T82

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7403218, 715786, -17291716, 742731, 12169158, -1752538, -3499132, 726751, 419638, -97302, -20812, 4457, 498, -82, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 - 82*x^13 + 498*x^12 + 4457*x^11 - 20812*x^10 - 97302*x^9 + 419638*x^8 + 726751*x^7 - 3499132*x^6 - 1752538*x^5 + 12169158*x^4 + 742731*x^3 - 17291716*x^2 + 715786*x + 7403218)
 
gp: K = bnfinit(x^15 - 4*x^14 - 82*x^13 + 498*x^12 + 4457*x^11 - 20812*x^10 - 97302*x^9 + 419638*x^8 + 726751*x^7 - 3499132*x^6 - 1752538*x^5 + 12169158*x^4 + 742731*x^3 - 17291716*x^2 + 715786*x + 7403218, 1)
 

Normalized defining polynomial

\( x^{15} - 4 x^{14} - 82 x^{13} + 498 x^{12} + 4457 x^{11} - 20812 x^{10} - 97302 x^{9} + 419638 x^{8} + 726751 x^{7} - 3499132 x^{6} - 1752538 x^{5} + 12169158 x^{4} + 742731 x^{3} - 17291716 x^{2} + 715786 x + 7403218 \)

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:  \(288430689287392159449927031844765696=2^{20}\cdot 37^{5}\cdot 43^{2}\cdot 139^{2}\cdot 317^{3}\cdot 1867^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $231.21$
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:  $2, 37, 43, 139, 317, 1867$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1884101965162260517857486811984475114646716} a^{14} - \frac{2472543392539425219336480493493855171881}{32484516640728629618232531241111639907702} a^{13} + \frac{204519635864938876446825840580476357223619}{1884101965162260517857486811984475114646716} a^{12} + \frac{224664327402463987136628432495713362550705}{942050982581130258928743405992237557323358} a^{11} - \frac{120764644150691835575236601566796430191615}{942050982581130258928743405992237557323358} a^{10} - \frac{4769933249496095599335609112943189964536}{471025491290565129464371702996118778661679} a^{9} - \frac{451523274070503553857239559582902664534825}{942050982581130258928743405992237557323358} a^{8} - \frac{96387451486906356077849443797374900637697}{942050982581130258928743405992237557323358} a^{7} + \frac{4325696630552072223922065989939199172907}{64969033281457259236465062482223279815404} a^{6} - \frac{228420457021368884381814308567446116040856}{471025491290565129464371702996118778661679} a^{5} + \frac{123299347208455851522564108364788563824595}{1884101965162260517857486811984475114646716} a^{4} + \frac{52322573175845600227036726163394264526750}{471025491290565129464371702996118778661679} a^{3} + \frac{72187102764408533048227774926101195225083}{942050982581130258928743405992237557323358} a^{2} + \frac{466924589780099666387126688667080448915059}{942050982581130258928743405992237557323358} a + \frac{40626675591167553112991577566776697822933}{471025491290565129464371702996118778661679}$

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:  \( 12436007749000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T82:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48000
The 65 conjugacy class representatives for [F(5)^3]S(3)=F(5)wrS(3) are not computed
Character table for [F(5)^3]S(3)=F(5)wrS(3) is not computed

Intermediate fields

3.3.148.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ R ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.18.78$x^{12} + 2 x^{11} + 2 x^{10} + 2 x^{8} + 2 x^{7} + 2 x^{2} - 2$$12$$1$$18$12T98$[4/3, 4/3, 5/3, 5/3, 2]_{3}^{2}$
37Data not computed
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.8.0.1$x^{8} - 3 x + 18$$1$$8$$0$$C_8$$[\ ]^{8}$
139Data not computed
317Data not computed
1867Data not computed