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Properties

Label 15.15.2375614325...7441.3
Degree $15$
Signature $[15, 0]$
Discriminant $11^{12}\cdot 31^{14}$
Root discriminant $167.90$
Ramified primes $11, 31$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_{15}$ (as 15T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2683361, 18560312, -29662843, 191261, 15651747, -3764830, -3121135, 1029158, 294231, -113742, -13489, 5937, 261, -138, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 138*x^13 + 261*x^12 + 5937*x^11 - 13489*x^10 - 113742*x^9 + 294231*x^8 + 1029158*x^7 - 3121135*x^6 - 3764830*x^5 + 15651747*x^4 + 191261*x^3 - 29662843*x^2 + 18560312*x + 2683361)
 
gp: K = bnfinit(x^15 - x^14 - 138*x^13 + 261*x^12 + 5937*x^11 - 13489*x^10 - 113742*x^9 + 294231*x^8 + 1029158*x^7 - 3121135*x^6 - 3764830*x^5 + 15651747*x^4 + 191261*x^3 - 29662843*x^2 + 18560312*x + 2683361, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 138 x^{13} + 261 x^{12} + 5937 x^{11} - 13489 x^{10} - 113742 x^{9} + 294231 x^{8} + 1029158 x^{7} - 3121135 x^{6} - 3764830 x^{5} + 15651747 x^{4} + 191261 x^{3} - 29662843 x^{2} + 18560312 x + 2683361 \)

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:  \(2375614325883809574306005975647441=11^{12}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $167.90$
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:  $11, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(341=11\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{341}(1,·)$, $\chi_{341}(67,·)$, $\chi_{341}(262,·)$, $\chi_{341}(38,·)$, $\chi_{341}(103,·)$, $\chi_{341}(9,·)$, $\chi_{341}(47,·)$, $\chi_{341}(80,·)$, $\chi_{341}(81,·)$, $\chi_{341}(82,·)$, $\chi_{341}(163,·)$, $\chi_{341}(245,·)$, $\chi_{341}(56,·)$, $\chi_{341}(159,·)$, $\chi_{341}(312,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{276248960266657575494685372195922746059267} a^{14} - \frac{85925854511237384478225500795741910312463}{276248960266657575494685372195922746059267} a^{13} + \frac{8373478925809533940534947718802773993868}{276248960266657575494685372195922746059267} a^{12} + \frac{136071231065739603175482383234441652836514}{276248960266657575494685372195922746059267} a^{11} - \frac{130839499445356603814820265083194449070931}{276248960266657575494685372195922746059267} a^{10} + \frac{97325609819606778653968903353297603201838}{276248960266657575494685372195922746059267} a^{9} - \frac{88657979988158372280863204852537817729354}{276248960266657575494685372195922746059267} a^{8} - \frac{2857492960402745451323648935792482385551}{276248960266657575494685372195922746059267} a^{7} + \frac{8840363361253377138133006869141507394868}{276248960266657575494685372195922746059267} a^{6} - \frac{52095461050954639163327317866065934251485}{276248960266657575494685372195922746059267} a^{5} + \frac{134384996912850782621500522554602504285559}{276248960266657575494685372195922746059267} a^{4} - \frac{132349794044341215870954439955226777989811}{276248960266657575494685372195922746059267} a^{3} + \frac{136876673469262378655053511368773674375586}{276248960266657575494685372195922746059267} a^{2} + \frac{112119690374223991240610795759346850154379}{276248960266657575494685372195922746059267} a + \frac{105798236941134950286741005438853317057886}{276248960266657575494685372195922746059267}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 93963096289.74506 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{15}$ (as 15T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 15
The 15 conjugacy class representatives for $C_{15}$
Character table for $C_{15}$

Intermediate fields

3.3.961.1, 5.5.13521270961.4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ $15$ $15$ $15$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{5}$ $15$ $15$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ R $15$ $15$ $15$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ $15$ $15$

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
$11$11.15.12.3$x^{15} - 22 x^{10} + 121 x^{5} - 11979$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$
$31$31.15.14.4$x^{15} + 10633$$15$$1$$14$$C_{15}$$[\ ]_{15}$