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Properties

Label 8.8.3038765625.1
Degree $8$
Signature $[8, 0]$
Discriminant $3038765625$
Root discriminant \(15.32\)
Ramified primes $3,5,7$
Class number $1$
Class group trivial
Galois group $C_4\times C_2$ (as 8T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 14*x^6 + 9*x^5 + 61*x^4 - 5*x^3 - 90*x^2 - 45*x - 5)
 
gp: K = bnfinit(y^8 - y^7 - 14*y^6 + 9*y^5 + 61*y^4 - 5*y^3 - 90*y^2 - 45*y - 5, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - x^7 - 14*x^6 + 9*x^5 + 61*x^4 - 5*x^3 - 90*x^2 - 45*x - 5);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - x^7 - 14*x^6 + 9*x^5 + 61*x^4 - 5*x^3 - 90*x^2 - 45*x - 5)
 

\( x^{8} - x^{7} - 14x^{6} + 9x^{5} + 61x^{4} - 5x^{3} - 90x^{2} - 45x - 5 \) 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:   \(3038765625\) \(\medspace = 3^{4}\cdot 5^{6}\cdot 7^{4}\) 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:  \(15.32\)
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:  $3^{1/2}5^{3/4}7^{1/2}\approx 15.322765339111537$
Ramified primes:   \(3\), \(5\), \(7\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(105=3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(97,·)$, $\chi_{105}(8,·)$, $\chi_{105}(41,·)$, $\chi_{105}(13,·)$, $\chi_{105}(104,·)$, $\chi_{105}(92,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{89}a^{7}+\frac{41}{89}a^{6}+\frac{17}{89}a^{5}+\frac{11}{89}a^{4}-\frac{11}{89}a^{3}-\frac{22}{89}a^{2}-\frac{35}{89}a-\frac{2}{89}$ 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:   $\frac{15}{89}a^{7}-\frac{8}{89}a^{6}-\frac{190}{89}a^{5}+\frac{76}{89}a^{4}+\frac{725}{89}a^{3}-\frac{63}{89}a^{2}-\frac{970}{89}a-\frac{208}{89}$, $\frac{4}{89}a^{7}-\frac{14}{89}a^{6}-\frac{21}{89}a^{5}+\frac{133}{89}a^{4}-\frac{133}{89}a^{3}-\frac{266}{89}a^{2}+\frac{483}{89}a+\frac{348}{89}$, $\frac{18}{89}a^{7}-\frac{63}{89}a^{6}-\frac{228}{89}a^{5}+\frac{643}{89}a^{4}+\frac{870}{89}a^{3}-\frac{1286}{89}a^{2}-\frac{1431}{89}a-\frac{303}{89}$, $\frac{34}{89}a^{7}-\frac{30}{89}a^{6}-\frac{401}{89}a^{5}+\frac{285}{89}a^{4}+\frac{1317}{89}a^{3}-\frac{303}{89}a^{2}-\frac{1457}{89}a-\frac{513}{89}$, $\frac{8}{89}a^{7}-\frac{28}{89}a^{6}-\frac{131}{89}a^{5}+\frac{266}{89}a^{4}+\frac{713}{89}a^{3}-\frac{354}{89}a^{2}-\frac{1437}{89}a-\frac{728}{89}$, $\frac{44}{89}a^{7}-\frac{65}{89}a^{6}-\frac{587}{89}a^{5}+\frac{662}{89}a^{4}+\frac{2364}{89}a^{3}-\frac{1235}{89}a^{2}-\frac{3320}{89}a-\frac{533}{89}$, $\frac{40}{89}a^{7}-\frac{51}{89}a^{6}-\frac{566}{89}a^{5}+\frac{529}{89}a^{4}+\frac{2497}{89}a^{3}-\frac{969}{89}a^{2}-\frac{3803}{89}a-\frac{792}{89}$ 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:  \( 112.572494501 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 112.572494501 \cdot 1}{2\cdot\sqrt{3038765625}}\cr\approx \mathstrut & 0.261392821698 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 14*x^6 + 9*x^5 + 61*x^4 - 5*x^3 - 90*x^2 - 45*x - 5)
 
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 - 14*x^6 + 9*x^5 + 61*x^4 - 5*x^3 - 90*x^2 - 45*x - 5, 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 - 14*x^6 + 9*x^5 + 61*x^4 - 5*x^3 - 90*x^2 - 45*x - 5);
 
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 - 14*x^6 + 9*x^5 + 61*x^4 - 5*x^3 - 90*x^2 - 45*x - 5);
 
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

$C_2\times C_4$ (as 8T2):

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)
 
An abelian group of order 8
The 8 conjugacy class representatives for $C_4\times C_2$
Character table for $C_4\times C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\zeta_{15})^+\), 4.4.6125.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{2}$ R R R ${\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.1.0.1}{1} }^{8}$

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 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$
\(7\) Copy content Toggle raw display 7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.105.2t1.a.a$1$ $ 3 \cdot 5 \cdot 7 $ \(\Q(\sqrt{105}) \) $C_2$ (as 2T1) $1$ $1$
* 1.15.4t1.a.a$1$ $ 3 \cdot 5 $ \(\Q(\zeta_{15})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.35.4t1.a.a$1$ $ 5 \cdot 7 $ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 1.21.2t1.a.a$1$ $ 3 \cdot 7 $ \(\Q(\sqrt{21}) \) $C_2$ (as 2T1) $1$ $1$
* 1.15.4t1.a.b$1$ $ 3 \cdot 5 $ \(\Q(\zeta_{15})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.35.4t1.a.b$1$ $ 5 \cdot 7 $ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$

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 *.