Properties

Label 9.1.72313663744.1
Degree $9$
Signature $[1, 4]$
Discriminant $72313663744$
Root discriminant \(16.09\)
Ramified primes see page
Class number $1$
Class group trivial
Galois group $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 4*x^7 + 28*x^3 + 26*x^2 + 9*x + 1)
 
gp: K = bnfinit(x^9 - x^8 - 4*x^7 + 28*x^3 + 26*x^2 + 9*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 9, 26, 28, 0, 0, 0, -4, -1, 1]);
 

\( x^{9} - x^{8} - 4x^{7} + 28x^{3} + 26x^{2} + 9x + 1 \) 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:  $[1, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(72313663744\) \(\medspace = 2^{8}\cdot 7^{10}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(16.09\)
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:   \(2\), \(7\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{758}a^{8}+\frac{130}{379}a^{7}-\frac{182}{379}a^{6}-\frac{127}{379}a^{5}-\frac{174}{379}a^{4}+\frac{66}{379}a^{3}+\frac{185}{379}a^{2}+\frac{165}{379}a-\frac{273}{758}$ 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

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $4$
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:   $a$, $\frac{1068}{379}a^{8}-\frac{1264}{379}a^{7}-\frac{4067}{379}a^{6}+\frac{850}{379}a^{5}-\frac{244}{379}a^{4}-\frac{12}{379}a^{3}+\frac{30183}{379}a^{2}+\frac{21952}{379}a+\frac{5193}{379}$, $\frac{1402}{379}a^{8}-\frac{1594}{379}a^{7}-\frac{5500}{379}a^{6}+\frac{910}{379}a^{5}+\frac{256}{379}a^{4}-\frac{267}{379}a^{3}+\frac{39305}{379}a^{2}+\frac{31358}{379}a+\frac{4971}{379}$, $\frac{647}{379}a^{8}-\frac{814}{379}a^{7}-\frac{2423}{379}a^{6}+\frac{527}{379}a^{5}-\frac{30}{379}a^{4}+\frac{129}{379}a^{3}+\frac{18433}{379}a^{2}+\frac{13019}{379}a+\frac{3015}{379}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 295.021698724 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{4}\cdot 295.021698724 \cdot 1}{2\sqrt{72313663744}}\approx 1.70987061986$

Galois group

$\PSL(2,8).C_3$ (as 9T32):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 1512
The 11 conjugacy class representatives for $\mathrm{P}\Gamma\mathrm{L}(2,8)$
Character table for $\mathrm{P}\Gamma\mathrm{L}(2,8)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 27 sibling: data not computed
Degree 28 sibling: data not computed
Degree 36 sibling: 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 R ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ R ${\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.9.0.1}{9} }$ ${\href{/padicField/19.9.0.1}{9} }$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.9.0.1}{9} }$ ${\href{/padicField/31.9.0.1}{9} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.3.0.1}{3} }^{3}$ ${\href{/padicField/43.9.0.1}{9} }$ ${\href{/padicField/47.9.0.1}{9} }$ ${\href{/padicField/53.9.0.1}{9} }$ ${\href{/padicField/59.9.0.1}{9} }$

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
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.8.8.13$x^{8} + 2 x + 2$$8$$1$$8$$C_2^3:(C_7: C_3)$$[8/7, 8/7, 8/7]_{7}^{3}$
\(7\) Copy content Toggle raw display 7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.7.10.5$x^{7} + 35 x^{4} + 7$$7$$1$$10$$F_7$$[5/3]_{3}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.7.3t1.a.a$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
1.7.3t1.a.b$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
7.72313663744.56.a.a$7$ $ 2^{8} \cdot 7^{10}$ 9.1.72313663744.1 $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $1$ $-1$
7.506195646208.168.a.a$7$ $ 2^{8} \cdot 7^{11}$ 9.1.72313663744.1 $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $0$ $-1$
7.506195646208.168.a.b$7$ $ 2^{8} \cdot 7^{11}$ 9.1.72313663744.1 $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $0$ $-1$
* 8.72313663744.9t32.a.a$8$ $ 2^{8} \cdot 7^{10}$ 9.1.72313663744.1 $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $1$ $0$
8.354...456.27t391.a.a$8$ $ 2^{8} \cdot 7^{12}$ 9.1.72313663744.1 $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $0$ $0$
8.354...456.27t391.a.b$8$ $ 2^{8} \cdot 7^{12}$ 9.1.72313663744.1 $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $0$ $0$
21.185...416.56.a.a$21$ $ 2^{24} \cdot 7^{32}$ 9.1.72313663744.1 $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $1$ $-3$
27.334...376.28t165.a.a$27$ $ 2^{30} \cdot 7^{42}$ 9.1.72313663744.1 $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $1$ $3$

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 *.