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Properties

Label 8.8.3341024655409.1
Degree $8$
Signature $[8, 0]$
Discriminant $3.341\times 10^{12}$
Root discriminant \(36.77\)
Ramified primes $7,73$
Class number $1$
Class group trivial
Galois group $\SL(2,3)$ (as 8T12)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 26*x^6 + 25*x^5 + 160*x^4 + 60*x^3 - 145*x^2 - 107*x - 13)
 
gp: K = bnfinit(y^8 - 2*y^7 - 26*y^6 + 25*y^5 + 160*y^4 + 60*y^3 - 145*y^2 - 107*y - 13, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^7 - 26*x^6 + 25*x^5 + 160*x^4 + 60*x^3 - 145*x^2 - 107*x - 13);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 2*x^7 - 26*x^6 + 25*x^5 + 160*x^4 + 60*x^3 - 145*x^2 - 107*x - 13)
 

\( x^{8} - 2x^{7} - 26x^{6} + 25x^{5} + 160x^{4} + 60x^{3} - 145x^{2} - 107x - 13 \) Copy content Toggle raw display

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

Invariants

Degree:  $8$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[8, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(3341024655409\) \(\medspace = 7^{6}\cdot 73^{4}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(36.77\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $7^{3/4}73^{2/3}\approx 75.16899954665448$
Ramified primes:   \(7\), \(73\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{24277}a^{7}+\frac{3757}{24277}a^{6}-\frac{607}{2207}a^{5}+\frac{3600}{24277}a^{4}+\frac{10271}{24277}a^{3}+\frac{8319}{24277}a^{2}+\frac{200}{2207}a-\frac{8764}{24277}$ Copy content Toggle raw display

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

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);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
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);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{13702}{24277}a^{7}-\frac{37380}{24277}a^{6}-\frac{29829}{2207}a^{5}+\frac{578984}{24277}a^{4}+\frac{1747417}{24277}a^{3}-\frac{406286}{24277}a^{2}-\frac{141942}{2207}a-\frac{398718}{24277}$, $\frac{1842}{2207}a^{7}-\frac{5172}{2207}a^{6}-\frac{43563}{2207}a^{5}+\frac{80824}{2207}a^{4}+\frac{225892}{2207}a^{3}-\frac{65813}{2207}a^{2}-\frac{196075}{2207}a-\frac{41016}{2207}$, $a+1$, $\frac{11157}{24277}a^{7}-\frac{33807}{24277}a^{6}-\frac{23293}{2207}a^{5}+\frac{545136}{24277}a^{4}+\frac{1244234}{24277}a^{3}-\frac{651367}{24277}a^{2}-\frac{94778}{2207}a-\frac{65023}{24277}$, $\frac{9105}{24277}a^{7}-\frac{23085}{24277}a^{6}-\frac{20270}{2207}a^{5}+\frac{343928}{24277}a^{4}+\frac{1240578}{24277}a^{3}-\frac{72576}{24277}a^{2}-\frac{103504}{2207}a-\frac{410430}{24277}$, $\frac{36152}{24277}a^{7}-\frac{103859}{24277}a^{6}-\frac{77308}{2207}a^{5}+\frac{1649039}{24277}a^{4}+\frac{4370337}{24277}a^{3}-\frac{1694378}{24277}a^{2}-\frac{352852}{2207}a-\frac{409710}{24277}$, $\frac{8045}{24277}a^{7}-\frac{24077}{24277}a^{6}-\frac{16880}{2207}a^{5}+\frac{387971}{24277}a^{4}+\frac{913813}{24277}a^{3}-\frac{466597}{24277}a^{2}-\frac{74941}{2207}a-\frac{103080}{24277}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 4505.33153315 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 4505.33153315 \cdot 1}{2\cdot\sqrt{3341024655409}}\cr\approx \mathstrut & 0.315498198833 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 26*x^6 + 25*x^5 + 160*x^4 + 60*x^3 - 145*x^2 - 107*x - 13)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^8 - 2*x^7 - 26*x^6 + 25*x^5 + 160*x^4 + 60*x^3 - 145*x^2 - 107*x - 13, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^8 - 2*x^7 - 26*x^6 + 25*x^5 + 160*x^4 + 60*x^3 - 145*x^2 - 107*x - 13);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 2*x^7 - 26*x^6 + 25*x^5 + 160*x^4 + 60*x^3 - 145*x^2 - 107*x - 13);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$\SL(2,3)$ (as 8T12):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 24
The 7 conjugacy class representatives for $\SL(2,3)$
Character table for $\SL(2,3)$

Intermediate fields

4.4.261121.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 24
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ ${\href{/padicField/3.4.0.1}{4} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ R ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display 7.8.6.1$x^{8} + 14 x^{4} - 245$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
\(73\) Copy content Toggle raw display 73.2.0.1$x^{2} + 70 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
73.6.4.1$x^{6} + 210 x^{5} + 14715 x^{4} + 345246 x^{3} + 88905 x^{2} + 1076160 x + 24967804$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.73.3t1.a.a$1$ $ 73 $ 3.3.5329.1 $C_3$ (as 3T1) $0$ $1$
1.73.3t1.a.b$1$ $ 73 $ 3.3.5329.1 $C_3$ (as 3T1) $0$ $1$
2.261121.24t7.a.a$2$ $ 7^{2} \cdot 73^{2}$ 8.8.3341024655409.1 $\SL(2,3)$ (as 8T12) $-1$ $2$
* 2.3577.8t12.a.a$2$ $ 7^{2} \cdot 73 $ 8.8.3341024655409.1 $\SL(2,3)$ (as 8T12) $0$ $2$
* 2.3577.8t12.a.b$2$ $ 7^{2} \cdot 73 $ 8.8.3341024655409.1 $\SL(2,3)$ (as 8T12) $0$ $2$
* 3.261121.4t4.a.a$3$ $ 7^{2} \cdot 73^{2}$ 4.4.261121.1 $A_4$ (as 4T4) $1$ $3$

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 *.