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Properties

Label 8.8.12782719397154721.1
Degree $8$
Signature $[8, 0]$
Discriminant $1.278\times 10^{16}$
Root discriminant \(103.12\)
Ramified primes $7,31$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^3:C_7$ (as 8T25)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 42*x^6 + 77*x^5 + 420*x^4 - 854*x^3 - 987*x^2 + 1577*x + 733)
 
gp: K = bnfinit(y^8 - y^7 - 42*y^6 + 77*y^5 + 420*y^4 - 854*y^3 - 987*y^2 + 1577*y + 733, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - x^7 - 42*x^6 + 77*x^5 + 420*x^4 - 854*x^3 - 987*x^2 + 1577*x + 733);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - x^7 - 42*x^6 + 77*x^5 + 420*x^4 - 854*x^3 - 987*x^2 + 1577*x + 733)
 

\( x^{8} - x^{7} - 42x^{6} + 77x^{5} + 420x^{4} - 854x^{3} - 987x^{2} + 1577x + 733 \) 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:   \(12782719397154721\) \(\medspace = 7^{12}\cdot 31^{4}\) 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:  \(103.12\)
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:  $7^{12/7}31^{1/2}\approx 156.46613112957195$
Ramified primes:   \(7\), \(31\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1081841}a^{7}+\frac{172360}{1081841}a^{6}-\frac{293783}{1081841}a^{5}-\frac{81740}{1081841}a^{4}+\frac{27623}{1081841}a^{3}-\frac{55192}{1081841}a^{2}-\frac{321386}{1081841}a+\frac{175795}{1081841}$ 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 $1$ (assuming GRH)

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:   $\frac{19847}{1081841}a^{7}+\frac{47678}{1081841}a^{6}-\frac{670052}{1081841}a^{5}-\frac{614121}{1081841}a^{4}+\frac{6231340}{1081841}a^{3}+\frac{509309}{1081841}a^{2}-\frac{11913657}{1081841}a-\frac{1015701}{1081841}$, $\frac{181077}{1081841}a^{7}+\frac{400711}{1081841}a^{6}-\frac{6467844}{1081841}a^{5}-\frac{7058305}{1081841}a^{4}+\frac{58958442}{1081841}a^{3}+\frac{36827968}{1081841}a^{2}-\frac{111569432}{1081841}a-\frac{69978034}{1081841}$, $\frac{337135}{1081841}a^{7}+\frac{744808}{1081841}a^{6}-\frac{11724724}{1081841}a^{5}-\frac{11579358}{1081841}a^{4}+\frac{102967672}{1081841}a^{3}+\frac{41620238}{1081841}a^{2}-\frac{188005930}{1081841}a-\frac{71249684}{1081841}$, $\frac{219949}{1081841}a^{7}+\frac{537318}{1081841}a^{6}-\frac{7568865}{1081841}a^{5}-\frac{9252250}{1081841}a^{4}+\frac{67106313}{1081841}a^{3}+\frac{39940770}{1081841}a^{2}-\frac{131941135}{1081841}a-\frac{55318617}{1081841}$, $\frac{451018}{1081841}a^{7}+\frac{695584}{1081841}a^{6}-\frac{17008552}{1081841}a^{5}-\frac{7888450}{1081841}a^{4}+\frac{164429090}{1081841}a^{3}+\frac{18967251}{1081841}a^{2}-\frac{364984980}{1081841}a-\frac{139893228}{1081841}$, $\frac{23368}{1081841}a^{7}+\frac{14437}{1081841}a^{6}-\frac{839999}{1081841}a^{5}+\frac{430886}{1081841}a^{4}+\frac{6126233}{1081841}a^{3}-\frac{3417707}{1081841}a^{2}-\frac{6498872}{1081841}a-\frac{1936399}{1081841}$, $\frac{11893}{56939}a^{7}+\frac{16541}{56939}a^{6}-\frac{468874}{56939}a^{5}-\frac{185090}{56939}a^{4}+\frac{4936002}{56939}a^{3}+\frac{620665}{56939}a^{2}-\frac{13707866}{56939}a-\frac{5479350}{56939}$ Copy content Toggle raw display (assuming GRH)
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:  \( 942691.948946 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 942691.948946 \cdot 1}{2\cdot\sqrt{12782719397154721}}\cr\approx \mathstrut & 1.06725485695 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 42*x^6 + 77*x^5 + 420*x^4 - 854*x^3 - 987*x^2 + 1577*x + 733)
 
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 - x^7 - 42*x^6 + 77*x^5 + 420*x^4 - 854*x^3 - 987*x^2 + 1577*x + 733, 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 - x^7 - 42*x^6 + 77*x^5 + 420*x^4 - 854*x^3 - 987*x^2 + 1577*x + 733);
 
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 - x^7 - 42*x^6 + 77*x^5 + 420*x^4 - 854*x^3 - 987*x^2 + 1577*x + 733);
 
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

$F_8$ (as 8T25):

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 56
The 8 conjugacy class representatives for $C_2^3:C_7$
Character table for $C_2^3:C_7$

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 14 sibling: deg 14
Degree 28 sibling: deg 28
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 ${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ R ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ R ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
7.7.12.1$x^{7} + 42 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
\(31\) Copy content Toggle raw display 31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$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$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.49.7t1.a.a$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
1.49.7t1.a.b$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
1.49.7t1.a.c$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
1.49.7t1.a.d$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
1.49.7t1.a.e$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
1.49.7t1.a.f$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
* 7.127...721.8t25.a.a$7$ $ 7^{12} \cdot 31^{4}$ 8.8.12782719397154721.1 $C_2^3:C_7$ (as 8T25) $1$ $7$

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 *.