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Properties

Label 15.15.2375614325...7441.2
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![-15313, -93070, 18479, 509073, 270601, -731294, -482477, 345794, 180337, -78278, -14853, 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 - 14853*x^10 - 78278*x^9 + 180337*x^8 + 345794*x^7 - 482477*x^6 - 731294*x^5 + 270601*x^4 + 509073*x^3 + 18479*x^2 - 93070*x - 15313)
 
gp: K = bnfinit(x^15 - x^14 - 138*x^13 + 261*x^12 + 5937*x^11 - 14853*x^10 - 78278*x^9 + 180337*x^8 + 345794*x^7 - 482477*x^6 - 731294*x^5 + 270601*x^4 + 509073*x^3 + 18479*x^2 - 93070*x - 15313, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 138 x^{13} + 261 x^{12} + 5937 x^{11} - 14853 x^{10} - 78278 x^{9} + 180337 x^{8} + 345794 x^{7} - 482477 x^{6} - 731294 x^{5} + 270601 x^{4} + 509073 x^{3} + 18479 x^{2} - 93070 x - 15313 \)

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$$\chi_{341}(1,·)$, $\chi_{341}(258,·)$, $\chi_{341}(67,·)$, $\chi_{341}(69,·)$, $\chi_{341}(70,·)$, $\chi_{341}(257,·)$, $\chi_{341}(328,·)$, $\chi_{341}(169,·)$, $\chi_{341}(295,·)$, $\chi_{341}(236,·)$, $\chi_{341}(113,·)$, $\chi_{341}(126,·)$, $\chi_{341}(56,·)$, $\chi_{341}(152,·)$, $\chi_{341}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{87} a^{10} - \frac{11}{87} a^{9} - \frac{5}{87} a^{8} - \frac{7}{87} a^{7} - \frac{17}{87} a^{6} - \frac{25}{87} a^{5} + \frac{1}{3} a^{4} + \frac{14}{87} a^{3} - \frac{5}{29} a^{2} - \frac{6}{29} a - \frac{4}{87}$, $\frac{1}{87} a^{11} - \frac{10}{87} a^{9} + \frac{25}{87} a^{8} + \frac{22}{87} a^{7} - \frac{3}{29} a^{6} + \frac{5}{29} a^{5} + \frac{43}{87} a^{4} - \frac{35}{87} a^{3} - \frac{38}{87} a^{2} + \frac{1}{87} a + \frac{14}{87}$, $\frac{1}{87} a^{12} + \frac{2}{87} a^{9} - \frac{28}{87} a^{8} + \frac{8}{87} a^{7} + \frac{19}{87} a^{6} - \frac{11}{29} a^{5} - \frac{2}{29} a^{4} + \frac{5}{29} a^{3} + \frac{25}{87} a^{2} + \frac{8}{87} a - \frac{40}{87}$, $\frac{1}{666507} a^{13} + \frac{3077}{666507} a^{12} - \frac{340}{222169} a^{11} - \frac{88}{22983} a^{10} + \frac{49727}{666507} a^{9} - \frac{172306}{666507} a^{8} - \frac{3227}{222169} a^{7} + \frac{182716}{666507} a^{6} - \frac{91}{14181} a^{5} - \frac{7700}{666507} a^{4} + \frac{87536}{666507} a^{3} + \frac{159658}{666507} a^{2} - \frac{110949}{222169} a - \frac{88777}{666507}$, $\frac{1}{141163374400471833523192083} a^{14} + \frac{16389405107572541174}{47054458133490611174397361} a^{13} - \frac{236009435283667550226616}{141163374400471833523192083} a^{12} - \frac{574916601490596125745785}{141163374400471833523192083} a^{11} + \frac{266644659914913867474943}{47054458133490611174397361} a^{10} + \frac{11551205349873240688330426}{141163374400471833523192083} a^{9} + \frac{67835534577058637875957418}{141163374400471833523192083} a^{8} + \frac{27884755724379009201567070}{141163374400471833523192083} a^{7} - \frac{65203592787771189685454335}{141163374400471833523192083} a^{6} + \frac{32925131617789059093772136}{141163374400471833523192083} a^{5} + \frac{3652290779744341695237167}{47054458133490611174397361} a^{4} + \frac{239098461630818724083009}{47054458133490611174397361} a^{3} - \frac{3081381242433137096809028}{141163374400471833523192083} a^{2} - \frac{64540105774182128513490325}{141163374400471833523192083} a - \frac{3715306991839965323515441}{141163374400471833523192083}$

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:  \( 228738890590.5845 \) (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, 5.5.13521270961.2

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ $15$ $15$ R $15$ $15$ $15$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{15}$ R $15$ $15$ $15$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ $15$

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$11.15.12.2$x^{15} - 121 x^{5} + 3993$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$
$31$31.15.14.7$x^{15} - 3647119$$15$$1$$14$$C_{15}$$[\ ]_{15}$