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Properties

Label 8.4.95738203125.2
Degree $8$
Signature $[4, 2]$
Discriminant $95738203125$
Root discriminant \(23.58\)
Ramified primes $3,5,41$
Class number $2$
Class group [2]
Galois group $C_8:C_2$ (as 8T7)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 2*x^6 - 28*x^5 - 35*x^4 + 193*x^3 - 133*x^2 + x + 331)
 
gp: K = bnfinit(y^8 - y^7 + 2*y^6 - 28*y^5 - 35*y^4 + 193*y^3 - 133*y^2 + y + 331, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - x^7 + 2*x^6 - 28*x^5 - 35*x^4 + 193*x^3 - 133*x^2 + x + 331);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - x^7 + 2*x^6 - 28*x^5 - 35*x^4 + 193*x^3 - 133*x^2 + x + 331)
 

\( x^{8} - x^{7} + 2x^{6} - 28x^{5} - 35x^{4} + 193x^{3} - 133x^{2} + x + 331 \) 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:  $[4, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(95738203125\) \(\medspace = 3^{6}\cdot 5^{7}\cdot 41^{2}\) 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:  \(23.58\)
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:  $3^{3/4}5^{7/8}41^{1/2}\approx 59.680385971954195$
Ramified primes:   \(3\), \(5\), \(41\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{201703949}a^{7}+\frac{22265957}{201703949}a^{6}+\frac{87953881}{201703949}a^{5}+\frac{94882487}{201703949}a^{4}+\frac{89908072}{201703949}a^{3}+\frac{50326508}{201703949}a^{2}+\frac{11113439}{201703949}a-\frac{47143382}{201703949}$ 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

$C_{2}$, which has order $2$

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:  $5$
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:   $\frac{1349}{3418711}a^{7}-\frac{18853}{3418711}a^{6}+\frac{1503}{3418711}a^{5}-\frac{64877}{3418711}a^{4}+\frac{378981}{3418711}a^{3}+\frac{1696254}{3418711}a^{2}-\frac{2437235}{3418711}a-\frac{1560296}{3418711}$, $\frac{538449}{201703949}a^{7}+\frac{1256082}{201703949}a^{6}+\frac{3973012}{201703949}a^{5}-\frac{11295598}{201703949}a^{4}-\frac{53339162}{201703949}a^{3}-\frac{62530211}{201703949}a^{2}+\frac{69061128}{201703949}a+\frac{336791081}{201703949}$, $\frac{1118324}{201703949}a^{7}-\frac{111931}{201703949}a^{6}+\frac{5285594}{201703949}a^{5}-\frac{25538897}{201703949}a^{4}-\frac{38305937}{201703949}a^{3}+\frac{90547071}{201703949}a^{2}-\frac{168373246}{201703949}a-\frac{197342148}{201703949}$, $\frac{1663286}{201703949}a^{7}-\frac{5817239}{201703949}a^{6}+\frac{13670399}{201703949}a^{5}-\frac{88096400}{201703949}a^{4}+\frac{134767839}{201703949}a^{3}+\frac{35646339}{201703949}a^{2}-\frac{129201602}{201703949}a+\frac{289716294}{201703949}$, $\frac{2769834}{201703949}a^{7}+\frac{5294898}{201703949}a^{6}+\frac{23831452}{201703949}a^{5}-\frac{9924749}{201703949}a^{4}-\frac{118669618}{201703949}a^{3}+\frac{87437364}{201703949}a^{2}-\frac{263569611}{201703949}a-\frac{441538917}{201703949}$ 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:  \( 152.651648266 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 152.651648266 \cdot 2}{2\cdot\sqrt{95738203125}}\cr\approx \mathstrut & 0.311629513227 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 2*x^6 - 28*x^5 - 35*x^4 + 193*x^3 - 133*x^2 + x + 331)
 
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 + 2*x^6 - 28*x^5 - 35*x^4 + 193*x^3 - 133*x^2 + x + 331, 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 + 2*x^6 - 28*x^5 - 35*x^4 + 193*x^3 - 133*x^2 + x + 331);
 
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 + 2*x^6 - 28*x^5 - 35*x^4 + 193*x^3 - 133*x^2 + x + 331);
 
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

$\OD_{16}$ (as 8T7):

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 16
The 10 conjugacy class representatives for $C_8:C_2$
Character table for $C_8:C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\)

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 16
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 ${\href{/padicField/2.8.0.1}{8} }$ R R ${\href{/padicField/7.8.0.1}{8} }$ ${\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }$ ${\href{/padicField/17.8.0.1}{8} }$ ${\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }$ ${\href{/padicField/29.1.0.1}{1} }^{8}$ ${\href{/padicField/31.1.0.1}{1} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }$ R ${\href{/padicField/43.8.0.1}{8} }$ ${\href{/padicField/47.8.0.1}{8} }$ ${\href{/padicField/53.8.0.1}{8} }$ ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(3\) Copy content Toggle raw display 3.8.6.3$x^{8} - 6 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
\(5\) Copy content Toggle raw display 5.8.7.1$x^{8} + 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
\(41\) Copy content Toggle raw display 41.4.2.2$x^{4} - 45592 x^{3} - 53825497 x^{2} - 1482642 x + 10086$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} + 23 x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.615.2t1.a.a$1$ $ 3 \cdot 5 \cdot 41 $ \(\Q(\sqrt{-615}) \) $C_2$ (as 2T1) $1$ $-1$
1.123.2t1.a.a$1$ $ 3 \cdot 41 $ \(\Q(\sqrt{-123}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 1.15.4t1.a.a$1$ $ 3 \cdot 5 $ \(\Q(\zeta_{15})^+\) $C_4$ (as 4T1) $0$ $1$
1.205.4t1.a.a$1$ $ 5 \cdot 41 $ 4.0.210125.1 $C_4$ (as 4T1) $0$ $-1$
1.205.4t1.a.b$1$ $ 5 \cdot 41 $ 4.0.210125.1 $C_4$ (as 4T1) $0$ $-1$
* 1.15.4t1.a.b$1$ $ 3 \cdot 5 $ \(\Q(\zeta_{15})^+\) $C_4$ (as 4T1) $0$ $1$
* 2.9225.8t7.b.a$2$ $ 3^{2} \cdot 5^{2} \cdot 41 $ 8.4.95738203125.2 $C_8:C_2$ (as 8T7) $0$ $0$
* 2.9225.8t7.b.b$2$ $ 3^{2} \cdot 5^{2} \cdot 41 $ 8.4.95738203125.2 $C_8:C_2$ (as 8T7) $0$ $0$

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 *.