Properties

Label 10.0.37078857421875.1
Degree $10$
Signature $[0, 5]$
Discriminant $-3.708\times 10^{13}$
Root discriminant \(22.75\)
Ramified primes $3,5$
Class number $11$
Class group [11]
Galois group $C_{10}$ (as 10T1)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 10*x^8 - 10*x^7 + 90*x^6 - 49*x^5 + 125*x^4 + 70*x^3 + 95*x^2 + 10*x + 1)
 
gp: K = bnfinit(y^10 + 10*y^8 - 10*y^7 + 90*y^6 - 49*y^5 + 125*y^4 + 70*y^3 + 95*y^2 + 10*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 + 10*x^8 - 10*x^7 + 90*x^6 - 49*x^5 + 125*x^4 + 70*x^3 + 95*x^2 + 10*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 + 10*x^8 - 10*x^7 + 90*x^6 - 49*x^5 + 125*x^4 + 70*x^3 + 95*x^2 + 10*x + 1)
 

\( x^{10} + 10x^{8} - 10x^{7} + 90x^{6} - 49x^{5} + 125x^{4} + 70x^{3} + 95x^{2} + 10x + 1 \) 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:   \(-37078857421875\) \(\medspace = -\,3^{5}\cdot 5^{16}\) 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:  \(22.75\)
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:  $3^{1/2}5^{8/5}\approx 22.746398023598275$
Ramified primes:   \(3\), \(5\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Gal(K/\Q) }$:  $10$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(75=3\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{75}(1,·)$, $\chi_{75}(71,·)$, $\chi_{75}(41,·)$, $\chi_{75}(11,·)$, $\chi_{75}(46,·)$, $\chi_{75}(16,·)$, $\chi_{75}(56,·)$, $\chi_{75}(26,·)$, $\chi_{75}(61,·)$, $\chi_{75}(31,·)$$\rbrace$
This is a CM field.
Reflex fields:  \(\Q(\sqrt{-3}) \), 10.0.37078857421875.1$^{15}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{1}{7}$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{2}$, $\frac{1}{1530907}a^{9}+\frac{897}{1530907}a^{8}-\frac{70185}{1530907}a^{7}+\frac{29933}{1530907}a^{6}-\frac{7176}{218701}a^{5}-\frac{94550}{218701}a^{4}+\frac{312591}{1530907}a^{3}+\frac{238216}{1530907}a^{2}+\frac{227671}{1530907}a+\frac{391565}{1530907}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $7$

Class group and class number

$C_{11}$, which has order $11$

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:   \( \frac{153600}{1530907} a^{9} - \frac{2430}{1530907} a^{8} + \frac{1543300}{1530907} a^{7} - \frac{1573230}{1530907} a^{6} + \frac{1987749}{218701} a^{5} - \frac{1133600}{218701} a^{4} + \frac{20043150}{1530907} a^{3} + \frac{8954835}{1530907} a^{2} + \frac{15284770}{1530907} a + \frac{1609000}{1530907} \)  (order $6$) 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:   $\frac{28218}{1530907}a^{9}-\frac{57770}{1530907}a^{8}+\frac{294627}{1530907}a^{7}-\frac{848672}{1530907}a^{6}+\frac{462160}{218701}a^{5}-\frac{953205}{218701}a^{4}+\frac{7261239}{1530907}a^{3}-\frac{5482373}{1530907}a^{2}-\frac{577700}{1530907}a-\frac{5060875}{1530907}$, $\frac{3441}{218701}a^{9}-\frac{6480}{218701}a^{8}+\frac{33048}{218701}a^{7}-\frac{102447}{218701}a^{6}+\frac{51840}{31243}a^{5}-\frac{106920}{31243}a^{4}+\frac{710216}{218701}a^{3}-\frac{614952}{218701}a^{2}-\frac{64800}{218701}a-\frac{166668}{218701}$, $\frac{23348}{1530907}a^{9}-\frac{52140}{1530907}a^{8}+\frac{265914}{1530907}a^{7}-\frac{748815}{1530907}a^{6}+\frac{417120}{218701}a^{5}-\frac{860310}{218701}a^{4}+\frac{8195534}{1530907}a^{3}-\frac{4948086}{1530907}a^{2}-\frac{521400}{1530907}a+\frac{2744830}{1530907}$, $\frac{210631}{1530907}a^{9}-\frac{21657}{1530907}a^{8}+\frac{2144370}{1530907}a^{7}-\frac{2300416}{1530907}a^{6}+\frac{2797668}{218701}a^{5}-\frac{1778897}{218701}a^{4}+\frac{30724805}{1530907}a^{3}+\frac{11569823}{1530907}a^{2}+\frac{23621839}{1530907}a+\frac{2486910}{1530907}$ 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:  \( 257.113789169 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 257.113789169 \cdot 11}{6\cdot\sqrt{37078857421875}}\cr\approx \mathstrut & 0.758058939924 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 + 10*x^8 - 10*x^7 + 90*x^6 - 49*x^5 + 125*x^4 + 70*x^3 + 95*x^2 + 10*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 + 10*x^8 - 10*x^7 + 90*x^6 - 49*x^5 + 125*x^4 + 70*x^3 + 95*x^2 + 10*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 + 10*x^8 - 10*x^7 + 90*x^6 - 49*x^5 + 125*x^4 + 70*x^3 + 95*x^2 + 10*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 + 10*x^8 - 10*x^7 + 90*x^6 - 49*x^5 + 125*x^4 + 70*x^3 + 95*x^2 + 10*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

$C_{10}$ (as 10T1):

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 cyclic group of order 10
The 10 conjugacy class representatives for $C_{10}$
Character table for $C_{10}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.390625.1

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 ${\href{/padicField/2.10.0.1}{10} }$ R R ${\href{/padicField/7.1.0.1}{1} }^{10}$ ${\href{/padicField/11.10.0.1}{10} }$ ${\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }$ ${\href{/padicField/29.10.0.1}{10} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }$ ${\href{/padicField/43.1.0.1}{1} }^{10}$ ${\href{/padicField/47.10.0.1}{10} }$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.10.0.1}{10} }$

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
\(3\) Copy content Toggle raw display 3.10.5.2$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(5\) Copy content Toggle raw display 5.10.16.7$x^{10} + 40 x^{9} + 400 x^{8} + 10 x^{5} + 200 x^{4} - 1225$$5$$2$$16$$C_{10}$$[2]^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.25.5t1.a.c$1$ $ 5^{2}$ 5.5.390625.1 $C_5$ (as 5T1) $0$ $1$
* 1.75.10t1.a.b$1$ $ 3 \cdot 5^{2}$ 10.0.37078857421875.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.25.5t1.a.a$1$ $ 5^{2}$ 5.5.390625.1 $C_5$ (as 5T1) $0$ $1$
* 1.75.10t1.a.c$1$ $ 3 \cdot 5^{2}$ 10.0.37078857421875.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.25.5t1.a.d$1$ $ 5^{2}$ 5.5.390625.1 $C_5$ (as 5T1) $0$ $1$
* 1.75.10t1.a.d$1$ $ 3 \cdot 5^{2}$ 10.0.37078857421875.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.25.5t1.a.b$1$ $ 5^{2}$ 5.5.390625.1 $C_5$ (as 5T1) $0$ $1$
* 1.75.10t1.a.a$1$ $ 3 \cdot 5^{2}$ 10.0.37078857421875.1 $C_{10}$ (as 10T1) $0$ $-1$

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 *.