Properties

Label 9.5.45487052759281.1
Degree $9$
Signature $[5, 2]$
Discriminant $4.549\times 10^{13}$
Root discriminant \(32.93\)
Ramified primes see page
Class number $3$
Class group $[3]$
Galois group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 7*x^7 - 7*x^6 - 7*x^5 + 84*x^4 + 35*x^3 - 280*x^2 + 28*x + 280)
 
gp: K = bnfinit(x^9 - 7*x^7 - 7*x^6 - 7*x^5 + 84*x^4 + 35*x^3 - 280*x^2 + 28*x + 280, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![280, 28, -280, 35, 84, -7, -7, -7, 0, 1]);
 

\( x^{9} - 7x^{7} - 7x^{6} - 7x^{5} + 84x^{4} + 35x^{3} - 280x^{2} + 28x + 280 \) Copy content Toggle raw display

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

Invariants

Degree:  $9$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[5, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(45487052759281\) \(\medspace = 7^{8}\cdot 53^{4}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(32.93\)
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:   \(7\), \(53\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Aut(K/\Q) }$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{6}a^{7}-\frac{1}{3}a^{6}-\frac{1}{6}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{4631724}a^{8}+\frac{153313}{2315862}a^{7}+\frac{15377}{514636}a^{6}-\frac{963277}{4631724}a^{5}-\frac{244643}{1543908}a^{4}-\frac{7749}{257318}a^{3}+\frac{531719}{4631724}a^{2}+\frac{364169}{771954}a-\frac{351383}{1157931}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{3}$, which has order $3$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $6$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $\frac{1426}{385977}a^{8}+\frac{2129}{771954}a^{7}-\frac{14488}{385977}a^{6}-\frac{3459}{257318}a^{5}-\frac{4079}{257318}a^{4}+\frac{46541}{257318}a^{3}+\frac{172466}{385977}a^{2}-\frac{1225327}{771954}a+\frac{115749}{128659}$, $\frac{25055}{4631724}a^{8}+\frac{14111}{2315862}a^{7}-\frac{60251}{1543908}a^{6}-\frac{535379}{4631724}a^{5}-\frac{215605}{1543908}a^{4}+\frac{123895}{257318}a^{3}+\frac{6013045}{4631724}a^{2}+\frac{272651}{771954}a-\frac{2081557}{1157931}$, $\frac{25055}{4631724}a^{8}+\frac{14111}{2315862}a^{7}-\frac{60251}{1543908}a^{6}-\frac{535379}{4631724}a^{5}-\frac{215605}{1543908}a^{4}+\frac{123895}{257318}a^{3}+\frac{6013045}{4631724}a^{2}-\frac{499303}{771954}a-\frac{2081557}{1157931}$, $\frac{608}{128659}a^{8}+\frac{1717}{128659}a^{7}-\frac{42}{128659}a^{6}-\frac{16648}{128659}a^{5}-\frac{39420}{128659}a^{4}-\frac{18775}{128659}a^{3}+\frac{93744}{128659}a^{2}+\frac{212996}{128659}a-\frac{139037}{128659}$, $\frac{10241}{4631724}a^{8}-\frac{76003}{2315862}a^{7}-\frac{2759}{514636}a^{6}+\frac{652363}{4631724}a^{5}+\frac{373721}{1543908}a^{4}+\frac{153753}{257318}a^{3}-\frac{10836593}{4631724}a^{2}-\frac{626999}{771954}a+\frac{4967969}{1157931}$, $\frac{112427}{4631724}a^{8}-\frac{27119}{1157931}a^{7}-\frac{131671}{1543908}a^{6}-\frac{144665}{4631724}a^{5}-\frac{529495}{1543908}a^{4}+\frac{233951}{128659}a^{3}-\frac{11353103}{4631724}a^{2}-\frac{228117}{128659}a+\frac{4441133}{1157931}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1696.63643085 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{5}\cdot(2\pi)^{2}\cdot 1696.63643085 \cdot 3}{2\sqrt{45487052759281}}\approx 0.476700780441$

Galois group

$C_3^3.S_4$ (as 9T30):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 648
The 14 conjugacy class representatives for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Character table for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$

Intermediate fields

3.3.2597.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 27 siblings: data not computed
Degree 36 siblings: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ R ${\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.3.0.1}{3} }^{3}$ ${\href{/padicField/17.9.0.1}{9} }$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.9.0.1}{9} }$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.3.0.1}{3} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.3.0.1}{3} }^{3}$ ${\href{/padicField/47.9.0.1}{9} }$ R ${\href{/padicField/59.3.0.1}{3} }^{3}$

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
\(7\) Copy content Toggle raw display 7.9.8.1$x^{9} + 14$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3}$
\(53\) Copy content Toggle raw display $\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.53.2t1.a.a$1$ $ 53 $ \(\Q(\sqrt{53}) \) $C_2$ (as 2T1) $1$ $1$
* 2.2597.3t2.a.a$2$ $ 7^{2} \cdot 53 $ 3.3.2597.1 $S_3$ (as 3T2) $1$ $2$
3.137641.6t8.a.a$3$ $ 7^{2} \cdot 53^{2}$ 4.0.2597.1 $S_4$ (as 4T5) $1$ $-1$
3.2597.4t5.a.a$3$ $ 7^{2} \cdot 53 $ 4.0.2597.1 $S_4$ (as 4T5) $1$ $-1$
6.17515230173.18t217.a.a$6$ $ 7^{6} \cdot 53^{3}$ 9.5.45487052759281.1 $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $2$
* 6.17515230173.9t30.a.a$6$ $ 7^{6} \cdot 53^{3}$ 9.5.45487052759281.1 $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $2$
6.17515230173.36t1121.a.a$6$ $ 7^{6} \cdot 53^{3}$ 9.5.45487052759281.1 $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $-2$
6.17515230173.36t1121.a.b$6$ $ 7^{6} \cdot 53^{3}$ 9.5.45487052759281.1 $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $-2$
8.928307199169.12t177.a.a$8$ $ 7^{6} \cdot 53^{4}$ 9.5.45487052759281.1 $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
8.454...281.12t177.a.a$8$ $ 7^{8} \cdot 53^{4}$ 9.5.45487052759281.1 $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
8.454...281.12t178.a.a$8$ $ 7^{8} \cdot 53^{4}$ 9.5.45487052759281.1 $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
12.331...413.36t1123.a.a$12$ $ 7^{10} \cdot 53^{7}$ 9.5.45487052759281.1 $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
12.118...757.18t218.a.a$12$ $ 7^{10} \cdot 53^{5}$ 9.5.45487052759281.1 $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$

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 *.