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Properties

Label 10.10.184...296.1
Degree $10$
Signature $[10, 0]$
Discriminant $1.847\times 10^{21}$
Root discriminant \(133.86\)
Ramified primes $2,3,421$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_{5}$ (as 10T7)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 54*x^8 - 14*x^7 + 819*x^6 + 174*x^5 - 4192*x^4 + 480*x^3 + 6708*x^2 - 856*x - 3336)
 
gp: K = bnfinit(y^10 - 54*y^8 - 14*y^7 + 819*y^6 + 174*y^5 - 4192*y^4 + 480*y^3 + 6708*y^2 - 856*y - 3336, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 54*x^8 - 14*x^7 + 819*x^6 + 174*x^5 - 4192*x^4 + 480*x^3 + 6708*x^2 - 856*x - 3336);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 54*x^8 - 14*x^7 + 819*x^6 + 174*x^5 - 4192*x^4 + 480*x^3 + 6708*x^2 - 856*x - 3336)
 

\( x^{10} - 54x^{8} - 14x^{7} + 819x^{6} + 174x^{5} - 4192x^{4} + 480x^{3} + 6708x^{2} - 856x - 3336 \) 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:  $[10, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1847300474809099063296\) \(\medspace = 2^{12}\cdot 3^{4}\cdot 421^{6}\) 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:  \(133.86\)
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:  $2^{3/2}3^{1/2}421^{2/3}\approx 275.18643248910337$
Ramified primes:   \(2\), \(3\), \(421\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{6}a^{6}-\frac{1}{3}a^{4}+\frac{1}{6}a^{2}$, $\frac{1}{18}a^{7}-\frac{1}{3}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}+\frac{4}{9}a-\frac{1}{3}$, $\frac{1}{108}a^{8}-\frac{1}{54}a^{7}-\frac{1}{18}a^{5}+\frac{1}{36}a^{4}+\frac{1}{9}a^{3}-\frac{17}{54}a^{2}-\frac{1}{27}a-\frac{2}{9}$, $\frac{1}{2464884}a^{9}+\frac{1873}{2464884}a^{8}-\frac{4915}{410814}a^{7}-\frac{7868}{205407}a^{6}+\frac{73495}{821628}a^{5}+\frac{124795}{821628}a^{4}-\frac{602279}{1232442}a^{3}-\frac{20776}{616221}a^{2}+\frac{33127}{68469}a-\frac{3818}{68469}$ 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:  $3$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:  $9$
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:   $\frac{151493}{2464884}a^{9}-\frac{97277}{1232442}a^{8}-\frac{148253}{45646}a^{7}+\frac{1360705}{410814}a^{6}+\frac{39124337}{821628}a^{5}-\frac{10276069}{205407}a^{4}-\frac{131400164}{616221}a^{3}+\frac{183118999}{616221}a^{2}+\frac{20304008}{205407}a-\frac{4335365}{22823}$, $\frac{83699}{2464884}a^{9}+\frac{65509}{2464884}a^{8}-\frac{2201861}{1232442}a^{7}-\frac{770939}{410814}a^{6}+\frac{20561161}{821628}a^{5}+\frac{20628871}{821628}a^{4}-\frac{129255181}{1232442}a^{3}-\frac{76298105}{1232442}a^{2}+\frac{64191847}{616221}a+\frac{13731215}{205407}$, $\frac{9641}{616221}a^{9}+\frac{4600}{616221}a^{8}-\frac{486262}{616221}a^{7}-\frac{276233}{410814}a^{6}+\frac{2056507}{205407}a^{5}+\frac{1951282}{205407}a^{4}-\frac{20725757}{616221}a^{3}-\frac{20480729}{1232442}a^{2}+\frac{17298130}{616221}a+\frac{1579073}{205407}$, $\frac{19893}{91292}a^{9}+\frac{181565}{410814}a^{8}-\frac{4620047}{410814}a^{7}-\frac{1784927}{68469}a^{6}+\frac{13266951}{91292}a^{5}+\frac{7957971}{22823}a^{4}-\frac{10999131}{22823}a^{3}-\frac{441151757}{410814}a^{2}+\frac{85416973}{205407}a+\frac{59342198}{68469}$, $\frac{28339}{2464884}a^{9}-\frac{18955}{616221}a^{8}-\frac{654115}{1232442}a^{7}+\frac{237688}{205407}a^{6}+\frac{4923239}{821628}a^{5}-\frac{4117193}{410814}a^{4}-\frac{10799998}{616221}a^{3}+\frac{30912899}{1232442}a^{2}+\frac{8718263}{616221}a-\frac{3498119}{205407}$, $\frac{79927}{2464884}a^{9}+\frac{186191}{1232442}a^{8}-\frac{37422}{22823}a^{7}-\frac{3447661}{410814}a^{6}+\frac{15342535}{821628}a^{5}+\frac{23860075}{205407}a^{4}-\frac{41482955}{1232442}a^{3}-\frac{268930927}{616221}a^{2}+\frac{4769566}{205407}a+\frac{8796145}{22823}$, $\frac{14765}{1232442}a^{9}+\frac{55207}{2464884}a^{8}-\frac{129773}{205407}a^{7}-\frac{532729}{410814}a^{6}+\frac{1774441}{205407}a^{5}+\frac{13409287}{821628}a^{4}-\frac{40393547}{1232442}a^{3}-\frac{26256667}{616221}a^{2}+\frac{2009308}{68469}a+\frac{2145442}{68469}$, $\frac{403253}{2464884}a^{9}+\frac{992719}{2464884}a^{8}-\frac{5136880}{616221}a^{7}-\frac{9399479}{410814}a^{6}+\frac{84306871}{821628}a^{5}+\frac{240756121}{821628}a^{4}-\frac{176910497}{616221}a^{3}-\frac{980551295}{1232442}a^{2}+\frac{166713325}{616221}a+\frac{109289756}{205407}$, $\frac{167233}{616221}a^{9}-\frac{506701}{2464884}a^{8}-\frac{17674255}{1232442}a^{7}+\frac{2917337}{410814}a^{6}+\frac{85721239}{410814}a^{5}-\frac{94599973}{821628}a^{4}-\frac{577697695}{616221}a^{3}+\frac{551304661}{616221}a^{2}+\frac{427337831}{616221}a-\frac{154412570}{205407}$ Copy content Toggle raw display (assuming GRH)
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:  \( 441037611.029 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 441037611.029 \cdot 1}{2\cdot\sqrt{1847300474809099063296}}\cr\approx \mathstrut & 5.25383969023 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 54*x^8 - 14*x^7 + 819*x^6 + 174*x^5 - 4192*x^4 + 480*x^3 + 6708*x^2 - 856*x - 3336)
 
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 - 54*x^8 - 14*x^7 + 819*x^6 + 174*x^5 - 4192*x^4 + 480*x^3 + 6708*x^2 - 856*x - 3336, 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 - 54*x^8 - 14*x^7 + 819*x^6 + 174*x^5 - 4192*x^4 + 480*x^3 + 6708*x^2 - 856*x - 3336);
 
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 - 54*x^8 - 14*x^7 + 819*x^6 + 174*x^5 - 4192*x^4 + 480*x^3 + 6708*x^2 - 856*x - 3336);
 
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: 5.5.102090816.1
Degree 6 sibling: 6.6.18094678318656.1
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: 5.5.102090816.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.3.0.1}{3} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.5.0.1}{5} }^{2}$ ${\href{/padicField/11.5.0.1}{5} }^{2}$ ${\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.5.0.1}{5} }^{2}$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.5.0.1}{5} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.5.0.1}{5} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2\) Copy content Toggle raw display 2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$$2$$3$$6$$A_4$$[2, 2]^{3}$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
\(421\) Copy content Toggle raw display $\Q_{421}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $3$$3$$1$$2$
Deg $3$$3$$1$$2$
Deg $3$$3$$1$$2$