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Properties

Label 15.15.3183001809...4049.1
Degree $15$
Signature $[15, 0]$
Discriminant $7^{10}\cdot 101^{12}$
Root discriminant $146.84$
Ramified primes $7, 101$
Class number $1$ (GRH)
Class group Trivial (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![229963, -2005313, 1384213, 5121648, -8403710, 3272277, 1891060, -2105658, 624063, 4666, -37014, 4862, 582, -131, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 131*x^13 + 582*x^12 + 4862*x^11 - 37014*x^10 + 4666*x^9 + 624063*x^8 - 2105658*x^7 + 1891060*x^6 + 3272277*x^5 - 8403710*x^4 + 5121648*x^3 + 1384213*x^2 - 2005313*x + 229963)
 
gp: K = bnfinit(x^15 - 2*x^14 - 131*x^13 + 582*x^12 + 4862*x^11 - 37014*x^10 + 4666*x^9 + 624063*x^8 - 2105658*x^7 + 1891060*x^6 + 3272277*x^5 - 8403710*x^4 + 5121648*x^3 + 1384213*x^2 - 2005313*x + 229963, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 131 x^{13} + 582 x^{12} + 4862 x^{11} - 37014 x^{10} + 4666 x^{9} + 624063 x^{8} - 2105658 x^{7} + 1891060 x^{6} + 3272277 x^{5} - 8403710 x^{4} + 5121648 x^{3} + 1384213 x^{2} - 2005313 x + 229963 \)

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:  \(318300180965960649704233197114049=7^{10}\cdot 101^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $146.84$
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:  $7, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(707=7\cdot 101\)
Dirichlet character group:    $\lbrace$$\chi_{707}(1,·)$, $\chi_{707}(387,·)$, $\chi_{707}(36,·)$, $\chi_{707}(102,·)$, $\chi_{707}(289,·)$, $\chi_{707}(137,·)$, $\chi_{707}(491,·)$, $\chi_{707}(589,·)$, $\chi_{707}(541,·)$, $\chi_{707}(592,·)$, $\chi_{707}(690,·)$, $\chi_{707}(499,·)$, $\chi_{707}(506,·)$, $\chi_{707}(701,·)$, $\chi_{707}(95,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} + \frac{1}{17} a^{11} - \frac{4}{17} a^{10} + \frac{8}{17} a^{9} - \frac{2}{17} a^{8} + \frac{2}{17} a^{7} + \frac{7}{17} a^{6} + \frac{2}{17} a^{5} + \frac{8}{17} a^{4} + \frac{6}{17} a^{3} - \frac{7}{17} a^{2} - \frac{3}{17} a - \frac{5}{17}$, $\frac{1}{697} a^{13} - \frac{16}{697} a^{12} + \frac{336}{697} a^{11} - \frac{179}{697} a^{10} - \frac{36}{697} a^{9} + \frac{189}{697} a^{8} + \frac{160}{697} a^{7} - \frac{304}{697} a^{6} + \frac{229}{697} a^{5} + \frac{142}{697} a^{4} + \frac{180}{697} a^{3} - \frac{139}{697} a^{2} + \frac{63}{697} a - \frac{17}{41}$, $\frac{1}{2737312400455839842137307302969343} a^{14} + \frac{1649659167455292845913172409604}{2737312400455839842137307302969343} a^{13} + \frac{43259800365359820972326095818882}{2737312400455839842137307302969343} a^{12} + \frac{1149013125699502827518371980113142}{2737312400455839842137307302969343} a^{11} - \frac{96668407606965666725410147886587}{2737312400455839842137307302969343} a^{10} + \frac{1032081151056489373110071090914587}{2737312400455839842137307302969343} a^{9} + \frac{560451049435184754062379812174656}{2737312400455839842137307302969343} a^{8} + \frac{733679772114180609028494303395871}{2737312400455839842137307302969343} a^{7} - \frac{1336073046350181907045825487454210}{2737312400455839842137307302969343} a^{6} - \frac{74616233869364574688521348434974}{2737312400455839842137307302969343} a^{5} - \frac{829224596279518752901730596488287}{2737312400455839842137307302969343} a^{4} + \frac{102010989449895656202212074801793}{2737312400455839842137307302969343} a^{3} - \frac{10529120486766409311304469791337}{66763717084288776637495300072423} a^{2} + \frac{1308885188668814407102394131770412}{2737312400455839842137307302969343} a - \frac{665860920156002335736478112723629}{2737312400455839842137307302969343}$

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:  \( 163490942894.50964 \) (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

\(\Q(\zeta_{7})^+\), 5.5.104060401.1

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 $15$ $15$ $15$ R $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ $15$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ $15$ $15$ $15$

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
$7$7.15.10.1$x^{15} + 4116 x^{6} - 2401 x^{3} + 1075648$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
$101$101.15.12.1$x^{15} + 5555 x^{10} + 6161404 x^{5} + 165931006351$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$