Properties

Label 15.1.845...879.1
Degree $15$
Signature $[1, 7]$
Discriminant $-8.450\times 10^{24}$
Root discriminant $45.90$
Ramified primes $3, 1213$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $D_{15}$ (as 15T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 29*x^13 - 119*x^12 + 396*x^11 - 1132*x^10 + 2906*x^9 - 6522*x^8 + 13105*x^7 - 24007*x^6 + 38721*x^5 - 52199*x^4 + 52599*x^3 - 37137*x^2 + 14859*x - 2421)
 
gp: K = bnfinit(x^15 - 6*x^14 + 29*x^13 - 119*x^12 + 396*x^11 - 1132*x^10 + 2906*x^9 - 6522*x^8 + 13105*x^7 - 24007*x^6 + 38721*x^5 - 52199*x^4 + 52599*x^3 - 37137*x^2 + 14859*x - 2421, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2421, 14859, -37137, 52599, -52199, 38721, -24007, 13105, -6522, 2906, -1132, 396, -119, 29, -6, 1]);
 

\(x^{15} - 6 x^{14} + 29 x^{13} - 119 x^{12} + 396 x^{11} - 1132 x^{10} + 2906 x^{9} - 6522 x^{8} + 13105 x^{7} - 24007 x^{6} + 38721 x^{5} - 52199 x^{4} + 52599 x^{3} - 37137 x^{2} + 14859 x - 2421\)  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:  $[1, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-8450344007000266933623879\)\(\medspace = -\,3^{7}\cdot 1213^{7}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $45.90$
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:  $3, 1213$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{5}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{6}$, $\frac{1}{81} a^{13} + \frac{4}{81} a^{12} - \frac{4}{81} a^{11} - \frac{1}{81} a^{10} + \frac{1}{27} a^{9} - \frac{1}{27} a^{8} + \frac{2}{81} a^{7} + \frac{8}{81} a^{6} + \frac{25}{81} a^{5} - \frac{32}{81} a^{4} - \frac{4}{9} a^{3} - \frac{1}{27} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{300775241343710331} a^{14} + \frac{552779285442629}{300775241343710331} a^{13} + \frac{549848504754818}{33419471260412259} a^{12} - \frac{27017080172585}{942869095121349} a^{11} - \frac{884867963771006}{27343203758519121} a^{10} + \frac{398355002683082}{33419471260412259} a^{9} - \frac{1654418639651207}{27343203758519121} a^{8} + \frac{1276475409501634}{300775241343710331} a^{7} + \frac{4296757354831004}{100258413781236777} a^{6} - \frac{16096503895896796}{300775241343710331} a^{5} - \frac{66919588537724816}{300775241343710331} a^{4} + \frac{10088668923281891}{100258413781236777} a^{3} + \frac{4988107917544460}{100258413781236777} a^{2} - \frac{2781404904652334}{11139823753470753} a - \frac{9300966731360576}{33419471260412259}$  Toggle raw display

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

Class group and class number

$C_{7}$, which has order $7$ (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:  $7$
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:  \( 4974627.527237572 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{7}\cdot 4974627.527237572 \cdot 7}{2\sqrt{8450344007000266933623879}}\approx 4.63106090571296$ (assuming GRH)

Galois group

$D_{15}$ (as 15T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.3639.1, 5.1.13242321.1

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

Sibling fields

Galois closure: Deg 30

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.5.0.1}{5} }^{3}$ R $15$ $15$ ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ $15$ $15$ ${\href{/padicField/19.5.0.1}{5} }^{3}$ $15$ ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ $15$ ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.3.0.1}{3} }^{5}$ ${\href{/padicField/43.3.0.1}{3} }^{5}$ $15$ ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.5.0.1}{5} }^{3}$

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

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
1213Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3639.2t1.a.a$1$ $ 3 \cdot 1213 $ \(\Q(\sqrt{-3639}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3639.3t2.a.a$2$ $ 3 \cdot 1213 $ 3.1.3639.1 $S_3$ (as 3T2) $1$ $0$
* 2.3639.5t2.a.b$2$ $ 3 \cdot 1213 $ 5.1.13242321.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.3639.5t2.a.a$2$ $ 3 \cdot 1213 $ 5.1.13242321.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.3639.15t2.a.a$2$ $ 3 \cdot 1213 $ 15.1.8450344007000266933623879.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.3639.15t2.a.b$2$ $ 3 \cdot 1213 $ 15.1.8450344007000266933623879.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.3639.15t2.a.c$2$ $ 3 \cdot 1213 $ 15.1.8450344007000266933623879.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.3639.15t2.a.d$2$ $ 3 \cdot 1213 $ 15.1.8450344007000266933623879.1 $D_{15}$ (as 15T2) $1$ $0$

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 *.