Properties

Label 15.15.109...121.1
Degree $15$
Signature $[15, 0]$
Discriminant $1.094\times 10^{22}$
Root discriminant $29.46$
Ramified primes $3, 11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{15}$ (as 15T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 27*x^13 - 4*x^12 + 252*x^11 + 60*x^10 - 976*x^9 - 288*x^8 + 1473*x^7 + 384*x^6 - 765*x^5 - 168*x^4 + 150*x^3 + 27*x^2 - 9*x - 1)
 
gp: K = bnfinit(x^15 - 27*x^13 - 4*x^12 + 252*x^11 + 60*x^10 - 976*x^9 - 288*x^8 + 1473*x^7 + 384*x^6 - 765*x^5 - 168*x^4 + 150*x^3 + 27*x^2 - 9*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -9, 27, 150, -168, -765, 384, 1473, -288, -976, 60, 252, -4, -27, 0, 1]);
 

\(x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} - 765 x^{5} - 168 x^{4} + 150 x^{3} + 27 x^{2} - 9 x - 1\)  Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $15$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[15, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(10943023107606534329121\)\(\medspace = 3^{20}\cdot 11^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $29.46$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $15$
This field is Galois and abelian over $\Q$.
Conductor:  \(99=3^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{99}(64,·)$, $\chi_{99}(1,·)$, $\chi_{99}(34,·)$, $\chi_{99}(67,·)$, $\chi_{99}(4,·)$, $\chi_{99}(37,·)$, $\chi_{99}(70,·)$, $\chi_{99}(97,·)$, $\chi_{99}(16,·)$, $\chi_{99}(49,·)$, $\chi_{99}(82,·)$, $\chi_{99}(25,·)$, $\chi_{99}(58,·)$, $\chi_{99}(91,·)$, $\chi_{99}(31,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{164276239249} a^{14} + \frac{32330953608}{164276239249} a^{13} + \frac{74618147066}{164276239249} a^{12} - \frac{7544382288}{164276239249} a^{11} - \frac{50914705859}{164276239249} a^{10} + \frac{3571539703}{164276239249} a^{9} - \frac{62434792324}{164276239249} a^{8} - \frac{80812000837}{164276239249} a^{7} - \frac{13866789831}{164276239249} a^{6} - \frac{48159141661}{164276239249} a^{5} + \frac{60633542357}{164276239249} a^{4} - \frac{855515036}{164276239249} a^{3} - \frac{76391229334}{164276239249} a^{2} + \frac{263409277}{164276239249} a + \frac{11630783081}{164276239249}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 967645.576239 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{15}\cdot(2\pi)^{0}\cdot 967645.576239 \cdot 1}{2\sqrt{10943023107606534329121}}\approx 0.151554067476$ (assuming GRH)

Galois group

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

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

Intermediate fields

\(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\)

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 $15$ R $15$ $15$ R $15$ ${\href{/padicField/17.5.0.1}{5} }^{3}$ ${\href{/padicField/19.5.0.1}{5} }^{3}$ ${\href{/padicField/23.3.0.1}{3} }^{5}$ $15$ $15$ ${\href{/padicField/37.5.0.1}{5} }^{3}$ $15$ ${\href{/padicField/43.3.0.1}{3} }^{5}$ $15$ ${\href{/padicField/53.5.0.1}{5} }^{3}$ $15$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
11Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.11.5t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.99.15t1.a.a$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
* 1.99.15t1.a.b$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
* 1.11.5t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.99.15t1.a.c$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
* 1.99.15t1.a.d$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
* 1.11.5t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.99.15t1.a.e$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
* 1.99.15t1.a.f$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
* 1.11.5t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.99.15t1.a.g$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
* 1.99.15t1.a.h$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.