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Properties

Label 10.0.246071287.1
Degree $10$
Signature $[0, 5]$
Discriminant $-246071287$
Root discriminant \(6.90\)
Ramified primes $7,11$
Class number $1$
Class group trivial
Galois group $D_5\times C_5$ (as 10T6)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 7*x^8 - 12*x^7 + 15*x^6 - 15*x^5 + 12*x^4 - 7*x^3 + 4*x^2 - 2*x + 1)
 
gp: K = bnfinit(y^10 - 3*y^9 + 7*y^8 - 12*y^7 + 15*y^6 - 15*y^5 + 12*y^4 - 7*y^3 + 4*y^2 - 2*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 3*x^9 + 7*x^8 - 12*x^7 + 15*x^6 - 15*x^5 + 12*x^4 - 7*x^3 + 4*x^2 - 2*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 3*x^9 + 7*x^8 - 12*x^7 + 15*x^6 - 15*x^5 + 12*x^4 - 7*x^3 + 4*x^2 - 2*x + 1)
 

\( x^{10} - 3x^{9} + 7x^{8} - 12x^{7} + 15x^{6} - 15x^{5} + 12x^{4} - 7x^{3} + 4x^{2} - 2x + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $10$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-246071287\) \(\medspace = -\,7^{5}\cdot 11^{4}\) 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:  \(6.90\)
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:  $7^{1/2}11^{4/5}\approx 18.016198912314337$
Ramified primes:   \(7\), \(11\) 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:  \(\Q(\sqrt{-7}) \)
$\card{ \Aut(K/\Q) }$:  $5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$ Copy content Toggle raw display

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

Monogenic:  Yes
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:  $4$
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:   $a^{9}-3a^{8}+6a^{7}-9a^{6}+9a^{5}-6a^{4}+3a^{3}$, $a^{9}-3a^{8}+7a^{7}-12a^{6}+14a^{5}-14a^{4}+10a^{3}-5a^{2}+3a-1$, $a^{8}-2a^{7}+4a^{6}-5a^{5}+4a^{4}-2a^{3}+a^{2}+1$, $a^{9}-2a^{8}+4a^{7}-5a^{6}+4a^{5}-2a^{4}+a^{2}$ 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:  \( 0.634803768223 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 0.634803768223 \cdot 1}{2\cdot\sqrt{246071287}}\cr\approx \mathstrut & 0.1981428346945 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 7*x^8 - 12*x^7 + 15*x^6 - 15*x^5 + 12*x^4 - 7*x^3 + 4*x^2 - 2*x + 1)
 
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^10 - 3*x^9 + 7*x^8 - 12*x^7 + 15*x^6 - 15*x^5 + 12*x^4 - 7*x^3 + 4*x^2 - 2*x + 1, 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^10 - 3*x^9 + 7*x^8 - 12*x^7 + 15*x^6 - 15*x^5 + 12*x^4 - 7*x^3 + 4*x^2 - 2*x + 1);
 
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^10 - 3*x^9 + 7*x^8 - 12*x^7 + 15*x^6 - 15*x^5 + 12*x^4 - 7*x^3 + 4*x^2 - 2*x + 1);
 
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

$C_5\times D_5$ (as 10T6):

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 50
The 20 conjugacy class representatives for $D_5\times C_5$
Character table for $D_5\times C_5$

Intermediate fields

\(\Q(\sqrt{-7}) \)

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

Degree 10 sibling: data not computed
Degree 25 sibling: data not computed
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 ${\href{/padicField/2.5.0.1}{5} }^{2}$ ${\href{/padicField/3.10.0.1}{10} }$ ${\href{/padicField/5.10.0.1}{10} }$ R R ${\href{/padicField/13.10.0.1}{10} }$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.10.0.1}{10} }$ ${\href{/padicField/23.5.0.1}{5} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }$ ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{5}$ ${\href{/padicField/41.10.0.1}{10} }$ ${\href{/padicField/43.5.0.1}{5} }^{2}$ ${\href{/padicField/47.10.0.1}{10} }$ ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{5}$ ${\href{/padicField/59.10.0.1}{10} }$

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 7.10.5.2$x^{10} + 35 x^{8} + 492 x^{6} + 8 x^{5} + 3360 x^{4} - 560 x^{3} + 11516 x^{2} + 1968 x + 17516$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(11\) Copy content Toggle raw display 11.5.0.1$x^{5} + 10 x^{2} + 9$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
1.11.5t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.11.5t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.77.10t1.a.d$1$ $ 7 \cdot 11 $ 10.0.3602729712967.1 $C_{10}$ (as 10T1) $0$ $-1$
1.11.5t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.77.10t1.a.a$1$ $ 7 \cdot 11 $ 10.0.3602729712967.1 $C_{10}$ (as 10T1) $0$ $-1$
1.77.10t1.a.c$1$ $ 7 \cdot 11 $ 10.0.3602729712967.1 $C_{10}$ (as 10T1) $0$ $-1$
1.11.5t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.77.10t1.a.b$1$ $ 7 \cdot 11 $ 10.0.3602729712967.1 $C_{10}$ (as 10T1) $0$ $-1$
* 2.77.10t6.b.b$2$ $ 7 \cdot 11 $ 10.0.246071287.1 $D_5\times C_5$ (as 10T6) $0$ $0$
2.847.5t2.a.a$2$ $ 7 \cdot 11^{2}$ 5.1.717409.1 $D_{5}$ (as 5T2) $1$ $0$
2.847.10t6.b.d$2$ $ 7 \cdot 11^{2}$ 10.0.246071287.1 $D_5\times C_5$ (as 10T6) $0$ $0$
2.847.10t6.b.b$2$ $ 7 \cdot 11^{2}$ 10.0.246071287.1 $D_5\times C_5$ (as 10T6) $0$ $0$
2.847.10t6.b.c$2$ $ 7 \cdot 11^{2}$ 10.0.246071287.1 $D_5\times C_5$ (as 10T6) $0$ $0$
* 2.77.10t6.b.c$2$ $ 7 \cdot 11 $ 10.0.246071287.1 $D_5\times C_5$ (as 10T6) $0$ $0$
* 2.77.10t6.b.a$2$ $ 7 \cdot 11 $ 10.0.246071287.1 $D_5\times C_5$ (as 10T6) $0$ $0$
2.847.5t2.a.b$2$ $ 7 \cdot 11^{2}$ 5.1.717409.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.77.10t6.b.d$2$ $ 7 \cdot 11 $ 10.0.246071287.1 $D_5\times C_5$ (as 10T6) $0$ $0$
2.847.10t6.b.a$2$ $ 7 \cdot 11^{2}$ 10.0.246071287.1 $D_5\times C_5$ (as 10T6) $0$ $0$

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 *.