Properties

Label 10.2.8123212515625.1
Degree $10$
Signature $[2, 4]$
Discriminant $8.123\times 10^{12}$
Root discriminant \(19.54\)
Ramified primes $5,151$
Class number $2$
Class group [2]
Galois group $A_{5}$ (as 10T7)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 2*x^8 - 5*x^7 + 23*x^6 - 19*x^5 - 45*x^4 + 109*x^3 - 104*x^2 + 67*x - 25)
 
gp: K = bnfinit(y^10 - 3*y^9 + 2*y^8 - 5*y^7 + 23*y^6 - 19*y^5 - 45*y^4 + 109*y^3 - 104*y^2 + 67*y - 25, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 3*x^9 + 2*x^8 - 5*x^7 + 23*x^6 - 19*x^5 - 45*x^4 + 109*x^3 - 104*x^2 + 67*x - 25);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 3*x^9 + 2*x^8 - 5*x^7 + 23*x^6 - 19*x^5 - 45*x^4 + 109*x^3 - 104*x^2 + 67*x - 25)
 

\( x^{10} - 3x^{9} + 2x^{8} - 5x^{7} + 23x^{6} - 19x^{5} - 45x^{4} + 109x^{3} - 104x^{2} + 67x - 25 \) 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:  $[2, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(8123212515625\) \(\medspace = 5^{6}\cdot 151^{4}\) 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:  \(19.54\)
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:  $5^{2/3}151^{1/2}\approx 35.93093151785668$
Ramified primes:   \(5\), \(151\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{165}a^{8}+\frac{23}{165}a^{7}+\frac{7}{55}a^{6}-\frac{16}{165}a^{5}-\frac{67}{165}a^{4}+\frac{26}{55}a^{3}-\frac{34}{165}a^{2}+\frac{43}{165}a+\frac{5}{11}$, $\frac{1}{825}a^{9}-\frac{41}{275}a^{7}-\frac{224}{825}a^{6}+\frac{301}{825}a^{5}+\frac{409}{825}a^{4}-\frac{68}{825}a^{3}+\frac{2}{5}a^{2}+\frac{62}{275}a-\frac{14}{33}$ 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:   $\frac{163}{825}a^{9}-\frac{76}{165}a^{8}+\frac{86}{825}a^{7}-\frac{767}{825}a^{6}+\frac{96}{25}a^{5}-\frac{366}{275}a^{4}-\frac{7724}{825}a^{3}+\frac{2617}{165}a^{2}-\frac{9122}{825}a+\frac{175}{33}$, $\frac{1}{11}a^{9}-\frac{37}{165}a^{8}-\frac{1}{165}a^{7}-\frac{67}{165}a^{6}+\frac{322}{165}a^{5}-\frac{7}{55}a^{4}-\frac{826}{165}a^{3}+\frac{1093}{165}a^{2}-\frac{76}{15}a+\frac{89}{33}$, $\frac{116}{825}a^{9}-\frac{41}{165}a^{8}-\frac{8}{825}a^{7}-\frac{589}{825}a^{6}+\frac{632}{275}a^{5}+\frac{43}{275}a^{4}-\frac{4903}{825}a^{3}+\frac{259}{33}a^{2}-\frac{4564}{825}a+\frac{104}{33}$, $\frac{14}{825}a^{9}-\frac{1}{11}a^{8}+\frac{128}{825}a^{7}-\frac{12}{275}a^{6}+\frac{464}{825}a^{5}-\frac{358}{275}a^{4}-\frac{159}{275}a^{3}+\frac{258}{55}a^{2}-\frac{4196}{825}a+\frac{52}{33}$, $\frac{34}{825}a^{9}-\frac{1}{15}a^{8}+\frac{53}{825}a^{7}-\frac{82}{275}a^{6}+\frac{389}{825}a^{5}-\frac{284}{825}a^{4}-\frac{184}{275}a^{3}+\frac{58}{15}a^{2}-\frac{3191}{825}a+\frac{41}{33}$ 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:  \( 269.012095622 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 269.012095622 \cdot 2}{2\cdot\sqrt{8123212515625}}\cr\approx \mathstrut & 0.588419917564 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 2*x^8 - 5*x^7 + 23*x^6 - 19*x^5 - 45*x^4 + 109*x^3 - 104*x^2 + 67*x - 25)
 
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 + 2*x^8 - 5*x^7 + 23*x^6 - 19*x^5 - 45*x^4 + 109*x^3 - 104*x^2 + 67*x - 25, 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 + 2*x^8 - 5*x^7 + 23*x^6 - 19*x^5 - 45*x^4 + 109*x^3 - 104*x^2 + 67*x - 25);
 
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 + 2*x^8 - 5*x^7 + 23*x^6 - 19*x^5 - 45*x^4 + 109*x^3 - 104*x^2 + 67*x - 25);
 
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

$A_5$ (as 10T7):

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 non-solvable group of order 60
The 5 conjugacy class representatives for $A_{5}$
Character table for $A_{5}$

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 5 sibling: 5.1.570025.1
Degree 6 sibling: 6.2.14250625.2
Degree 12 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 sibling: data not computed
Degree 30 sibling: data not computed
Minimal sibling: 5.1.570025.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.5.0.1}{5} }^{2}$ ${\href{/padicField/3.3.0.1}{3} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }$ R ${\href{/padicField/7.5.0.1}{5} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.5.0.1}{5} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.3.2.1$x^{3} + 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(151\) Copy content Toggle raw display $\Q_{151}$$x + 145$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 145$$1$$1$$0$Trivial$[\ ]$
151.2.1.1$x^{2} + 453$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.1$x^{2} + 453$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.1$x^{2} + 453$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.1$x^{2} + 453$$2$$1$$1$$C_2$$[\ ]_{2}$