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Properties

Label 15.15.5720845891...8729.1
Degree $15$
Signature $[15, 0]$
Discriminant $11^{12}\cdot 67^{10}$
Root discriminant $112.33$
Ramified primes $11, 67$
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![-4845499, -4647121, 17818309, 13947752, -9635844, -8394345, 1155822, 1590440, -22067, -136528, -2956, 5870, 152, -123, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 123*x^13 + 152*x^12 + 5870*x^11 - 2956*x^10 - 136528*x^9 - 22067*x^8 + 1590440*x^7 + 1155822*x^6 - 8394345*x^5 - 9635844*x^4 + 13947752*x^3 + 17818309*x^2 - 4647121*x - 4845499)
 
gp: K = bnfinit(x^15 - 2*x^14 - 123*x^13 + 152*x^12 + 5870*x^11 - 2956*x^10 - 136528*x^9 - 22067*x^8 + 1590440*x^7 + 1155822*x^6 - 8394345*x^5 - 9635844*x^4 + 13947752*x^3 + 17818309*x^2 - 4647121*x - 4845499, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 123 x^{13} + 152 x^{12} + 5870 x^{11} - 2956 x^{10} - 136528 x^{9} - 22067 x^{8} + 1590440 x^{7} + 1155822 x^{6} - 8394345 x^{5} - 9635844 x^{4} + 13947752 x^{3} + 17818309 x^{2} - 4647121 x - 4845499 \)

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:  \(5720845891965056149406296828729=11^{12}\cdot 67^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.33$
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:  $11, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(737=11\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{737}(1,·)$, $\chi_{737}(707,·)$, $\chi_{737}(37,·)$, $\chi_{737}(135,·)$, $\chi_{737}(104,·)$, $\chi_{737}(202,·)$, $\chi_{737}(364,·)$, $\chi_{737}(269,·)$, $\chi_{737}(498,·)$, $\chi_{737}(163,·)$, $\chi_{737}(372,·)$, $\chi_{737}(565,·)$, $\chi_{737}(632,·)$, $\chi_{737}(537,·)$, $\chi_{737}(573,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{11481} a^{13} - \frac{1558}{11481} a^{12} + \frac{502}{3827} a^{11} - \frac{640}{11481} a^{10} + \frac{1445}{11481} a^{9} + \frac{4280}{11481} a^{8} - \frac{4547}{11481} a^{7} - \frac{4502}{11481} a^{6} - \frac{1135}{3827} a^{5} + \frac{59}{11481} a^{4} + \frac{1699}{11481} a^{3} - \frac{790}{3827} a^{2} + \frac{441}{3827} a + \frac{1663}{3827}$, $\frac{1}{8333731061413040248364743841825934698067} a^{14} - \frac{26103286598070213814154026457916796}{8333731061413040248364743841825934698067} a^{13} - \frac{792834257909521093277345351796739563461}{8333731061413040248364743841825934698067} a^{12} - \frac{1357683583125600017977117996432911508355}{8333731061413040248364743841825934698067} a^{11} - \frac{1348127339036785098316046296647752975498}{8333731061413040248364743841825934698067} a^{10} + \frac{1305033336423274730990903293613008361363}{8333731061413040248364743841825934698067} a^{9} - \frac{1216358989801822267971045651528573472001}{2777910353804346749454914613941978232689} a^{8} - \frac{4062136448903759168495018174253458398664}{8333731061413040248364743841825934698067} a^{7} + \frac{3062493060017725701116768598009485655809}{8333731061413040248364743841825934698067} a^{6} - \frac{1473969502603099741674112744708737302675}{8333731061413040248364743841825934698067} a^{5} + \frac{940100341252318561146990441258388599363}{2777910353804346749454914613941978232689} a^{4} - \frac{175684087278837301571142458801655198698}{8333731061413040248364743841825934698067} a^{3} + \frac{3293913221866384364028349875457779384058}{8333731061413040248364743841825934698067} a^{2} + \frac{227685281704555796779774206329849459695}{8333731061413040248364743841825934698067} a + \frac{868649342749065498555478159700845295417}{8333731061413040248364743841825934698067}$

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:  \( 35980252831.5571 \) (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

3.3.4489.1, \(\Q(\zeta_{11})^+\)

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 $15$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ $15$ R $15$ $15$ $15$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{15}$ $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$11$11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$
$67$67.3.2.1$x^{3} - 67$$3$$1$$2$$C_3$$[\ ]_{3}$
67.3.2.1$x^{3} - 67$$3$$1$$2$$C_3$$[\ ]_{3}$
67.3.2.1$x^{3} - 67$$3$$1$$2$$C_3$$[\ ]_{3}$
67.3.2.1$x^{3} - 67$$3$$1$$2$$C_3$$[\ ]_{3}$
67.3.2.1$x^{3} - 67$$3$$1$$2$$C_3$$[\ ]_{3}$