Properties

Label 8.0.199390751295241.2
Degree $8$
Signature $[0, 4]$
Discriminant $1.994\times 10^{14}$
Root discriminant \(61.30\)
Ramified primes $11,103$
Class number $12$
Class group [12]
Galois group $\PSL(2,7)$ (as 8T37)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 29*x^6 - 113*x^5 + 383*x^4 - 1100*x^3 + 2424*x^2 - 3272*x + 5488)
 
gp: K = bnfinit(y^8 - 3*y^7 + 29*y^6 - 113*y^5 + 383*y^4 - 1100*y^3 + 2424*y^2 - 3272*y + 5488, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 3*x^7 + 29*x^6 - 113*x^5 + 383*x^4 - 1100*x^3 + 2424*x^2 - 3272*x + 5488);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 3*x^7 + 29*x^6 - 113*x^5 + 383*x^4 - 1100*x^3 + 2424*x^2 - 3272*x + 5488)
 

\( x^{8} - 3x^{7} + 29x^{6} - 113x^{5} + 383x^{4} - 1100x^{3} + 2424x^{2} - 3272x + 5488 \) 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:  $[0, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(199390751295241\) \(\medspace = 11^{6}\cdot 103^{4}\) 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:  \(61.30\)
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:  $11^{6/7}103^{1/2}\approx 79.25759937842756$
Ramified primes:   \(11\), \(103\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{11}a^{4}+\frac{3}{11}a^{3}-\frac{5}{11}a^{2}-\frac{5}{11}a-\frac{4}{11}$, $\frac{1}{22}a^{5}-\frac{1}{22}a^{4}+\frac{5}{22}a^{3}-\frac{7}{22}a^{2}+\frac{5}{22}a-\frac{3}{11}$, $\frac{1}{484}a^{6}-\frac{1}{44}a^{5}+\frac{13}{484}a^{4}-\frac{41}{484}a^{3}-\frac{157}{484}a^{2}+\frac{16}{121}a-\frac{38}{121}$, $\frac{1}{968}a^{7}-\frac{1}{968}a^{6}-\frac{9}{968}a^{5}+\frac{1}{968}a^{4}+\frac{357}{968}a^{3}+\frac{149}{484}a^{2}-\frac{5}{121}a-\frac{14}{121}$ 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

$C_{12}$, which has order $12$

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:  $3$
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:   $\frac{45}{968}a^{7}-\frac{63}{968}a^{6}+\frac{717}{968}a^{5}-\frac{2257}{968}a^{4}+\frac{2503}{968}a^{3}+2a^{2}-\frac{2509}{242}a+\frac{8545}{121}$, $\frac{1}{22}a^{7}-\frac{157}{484}a^{6}+\frac{71}{44}a^{5}-\frac{4043}{484}a^{4}+\frac{13543}{484}a^{3}-\frac{31583}{484}a^{2}+\frac{29087}{242}a-\frac{16474}{121}$, $\frac{225}{484}a^{7}-\frac{67}{242}a^{6}+\frac{965}{121}a^{5}-\frac{463}{22}a^{4}+\frac{2445}{121}a^{3}-\frac{16911}{484}a^{2}-\frac{1401}{242}a+\frac{105984}{121}$ 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:  \( 2081.22156701 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 2081.22156701 \cdot 12}{2\cdot\sqrt{199390751295241}}\cr\approx \mathstrut & 1.37827708793 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 29*x^6 - 113*x^5 + 383*x^4 - 1100*x^3 + 2424*x^2 - 3272*x + 5488)
 
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 - 3*x^7 + 29*x^6 - 113*x^5 + 383*x^4 - 1100*x^3 + 2424*x^2 - 3272*x + 5488, 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 - 3*x^7 + 29*x^6 - 113*x^5 + 383*x^4 - 1100*x^3 + 2424*x^2 - 3272*x + 5488);
 
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 - 3*x^7 + 29*x^6 - 113*x^5 + 383*x^4 - 1100*x^3 + 2424*x^2 - 3272*x + 5488);
 
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

$\GL(3,2)$ (as 8T37):

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 non-solvable group of order 168
The 6 conjugacy class representatives for $\PSL(2,7)$
Character table for $\PSL(2,7)$

Intermediate fields

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

Degree 7 siblings: 7.3.18794490649.1, 7.3.18794490649.2
Degree 14 siblings: deg 14, deg 14
Degree 21 sibling: deg 21
Degree 24 sibling: deg 24
Degree 28 sibling: deg 28
Degree 42 siblings: deg 42, some data not computed
Minimal sibling: 7.3.18794490649.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{2}$ ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ R ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$[\ ]$
11.7.6.1$x^{7} + 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
\(103\) Copy content Toggle raw display 103.2.1.2$x^{2} + 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 103$$2$$1$$1$$C_2$$[\ ]_{2}$