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Properties

Label 10.10.270...896.1
Degree $10$
Signature $[10, 0]$
Discriminant $2.701\times 10^{19}$
Root discriminant \(87.73\)
Ramified primes $2,11,29,113$
Class number $1$
Class group trivial
Galois group $C_2^4 : A_5$ (as 10T34)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 - 74*x^8 + 124*x^7 + 1446*x^6 - 3368*x^5 - 4389*x^4 + 9105*x^3 + 4770*x^2 - 2392*x + 146)
 
gp: K = bnfinit(y^10 - y^9 - 74*y^8 + 124*y^7 + 1446*y^6 - 3368*y^5 - 4389*y^4 + 9105*y^3 + 4770*y^2 - 2392*y + 146, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - x^9 - 74*x^8 + 124*x^7 + 1446*x^6 - 3368*x^5 - 4389*x^4 + 9105*x^3 + 4770*x^2 - 2392*x + 146);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - x^9 - 74*x^8 + 124*x^7 + 1446*x^6 - 3368*x^5 - 4389*x^4 + 9105*x^3 + 4770*x^2 - 2392*x + 146)
 

\( x^{10} - x^{9} - 74x^{8} + 124x^{7} + 1446x^{6} - 3368x^{5} - 4389x^{4} + 9105x^{3} + 4770x^{2} - 2392x + 146 \) 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:   \(27014472322230266896\) \(\medspace = 2^{4}\cdot 11^{4}\cdot 29^{4}\cdot 113^{4}\) 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:  \(87.73\)
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:  $2^{2/3}11^{2/3}29^{1/2}113^{2/3}\approx 988.2490718988355$
Ramified primes:   \(2\), \(11\), \(29\), \(113\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{5}$, $\frac{1}{32764048381432}a^{9}-\frac{697041614389}{16382024190716}a^{8}+\frac{498722138041}{8191012095358}a^{7}-\frac{710016782174}{4095506047679}a^{6}+\frac{3161719776657}{16382024190716}a^{5}-\frac{6006090300531}{16382024190716}a^{4}-\frac{8737905320775}{32764048381432}a^{3}+\frac{3776376853493}{8191012095358}a^{2}-\frac{6721793473917}{16382024190716}a-\frac{1289336433323}{16382024190716}$ 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:  $2$

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:  $9$
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:   $\frac{13363293937}{8191012095358}a^{9}-\frac{14464937623}{4095506047679}a^{8}-\frac{490894753298}{4095506047679}a^{7}+\frac{2788268653951}{8191012095358}a^{6}+\frac{8948779320626}{4095506047679}a^{5}-\frac{33332412387049}{4095506047679}a^{4}-\frac{6095987941086}{4095506047679}a^{3}+\frac{92031868051957}{4095506047679}a^{2}-\frac{50960110325924}{4095506047679}a+\frac{3990092349287}{4095506047679}$, $\frac{3335334171}{4095506047679}a^{9}-\frac{1144063257}{4095506047679}a^{8}-\frac{241324380143}{4095506047679}a^{7}+\frac{526882405847}{8191012095358}a^{6}+\frac{4638483996672}{4095506047679}a^{5}-\frac{8438462745097}{4095506047679}a^{4}-\frac{30990622545619}{8191012095358}a^{3}+\frac{21970308557431}{4095506047679}a^{2}+\frac{18161678731937}{4095506047679}a-\frac{889527618074}{4095506047679}$, $\frac{7738074061}{16382024190716}a^{9}+\frac{4923607201}{8191012095358}a^{8}-\frac{139720802775}{4095506047679}a^{7}-\frac{65547641130}{4095506047679}a^{6}+\frac{5714177484751}{8191012095358}a^{5}-\frac{1952543675571}{8191012095358}a^{4}-\frac{64431798039491}{16382024190716}a^{3}+\frac{2304590604251}{4095506047679}a^{2}+\frac{75107692361291}{8191012095358}a-\frac{20227487151445}{8191012095358}$, $\frac{21158677417}{16382024190716}a^{9}+\frac{6301998033}{8191012095358}a^{8}-\frac{382286063384}{4095506047679}a^{7}+\frac{96597169897}{8191012095358}a^{6}+\frac{14873716638459}{8191012095358}a^{5}-\frac{11935567908167}{8191012095358}a^{4}-\frac{108659262230149}{16382024190716}a^{3}+\frac{1385803113374}{4095506047679}a^{2}+\frac{11616773499765}{8191012095358}a-\frac{1632419462779}{8191012095358}$, $\frac{1075075427}{32764048381432}a^{9}+\frac{38747653807}{16382024190716}a^{8}-\frac{4854948196}{4095506047679}a^{7}-\frac{1296956621171}{8191012095358}a^{6}+\frac{1805206821309}{16382024190716}a^{5}+\frac{42363150809125}{16382024190716}a^{4}-\frac{136364077661937}{32764048381432}a^{3}-\frac{20705193976827}{8191012095358}a^{2}+\frac{32460357159933}{16382024190716}a-\frac{21812511457}{16382024190716}$, $\frac{464287782743}{16382024190716}a^{9}+\frac{336459795076}{4095506047679}a^{8}-\frac{7484438568287}{4095506047679}a^{7}-\frac{30239025217041}{8191012095358}a^{6}+\frac{122679759982109}{4095506047679}a^{5}+\frac{193146085186681}{8191012095358}a^{4}-\frac{14\!\cdots\!83}{16382024190716}a^{3}-\frac{239946202846873}{4095506047679}a^{2}+\frac{263565140593679}{8191012095358}a-\frac{23169932462441}{8191012095358}$, $\frac{41266491351}{16382024190716}a^{9}+\frac{46666784745}{4095506047679}a^{8}-\frac{1262094393261}{8191012095358}a^{7}-\frac{2569801224123}{4095506047679}a^{6}+\frac{8596644702628}{4095506047679}a^{5}+\frac{27781957586305}{4095506047679}a^{4}-\frac{68297205506233}{16382024190716}a^{3}-\frac{71559454437424}{4095506047679}a^{2}-\frac{59332569059793}{8191012095358}a+\frac{2994675020205}{8191012095358}$, $\frac{1400319064883}{32764048381432}a^{9}+\frac{1650084026959}{16382024190716}a^{8}-\frac{22776333763317}{8191012095358}a^{7}-\frac{32244588433121}{8191012095358}a^{6}+\frac{748902790527757}{16382024190716}a^{5}+\frac{96897385485359}{16382024190716}a^{4}-\frac{39\!\cdots\!77}{32764048381432}a^{3}-\frac{153259130600127}{8191012095358}a^{2}+\frac{495893951891377}{16382024190716}a-\frac{33948819949665}{16382024190716}$, $\frac{14300243525}{16382024190716}a^{9}-\frac{13558914043}{8191012095358}a^{8}-\frac{460460821201}{8191012095358}a^{7}+\frac{1510553280277}{8191012095358}a^{6}+\frac{5195663671653}{8191012095358}a^{5}-\frac{18690835369173}{4095506047679}a^{4}+\frac{125925801481835}{16382024190716}a^{3}+\frac{30222080548218}{4095506047679}a^{2}-\frac{182108977067905}{8191012095358}a+\frac{16439533586437}{8191012095358}$ 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:  \( 22394633.6643 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 22394633.6643 \cdot 1}{2\cdot\sqrt{27014472322230266896}}\cr\approx \mathstrut & 2.20605166438 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 - 74*x^8 + 124*x^7 + 1446*x^6 - 3368*x^5 - 4389*x^4 + 9105*x^3 + 4770*x^2 - 2392*x + 146)
 
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 - x^9 - 74*x^8 + 124*x^7 + 1446*x^6 - 3368*x^5 - 4389*x^4 + 9105*x^3 + 4770*x^2 - 2392*x + 146, 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 - x^9 - 74*x^8 + 124*x^7 + 1446*x^6 - 3368*x^5 - 4389*x^4 + 9105*x^3 + 4770*x^2 - 2392*x + 146);
 
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 - x^9 - 74*x^8 + 124*x^7 + 1446*x^6 - 3368*x^5 - 4389*x^4 + 9105*x^3 + 4770*x^2 - 2392*x + 146);
 
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

$C_2^4:A_5$ (as 10T34):

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 960
The 12 conjugacy class representatives for $C_2^4 : A_5$
Character table for $C_2^4 : A_5$

Intermediate fields

5.5.6180196.1

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]
 

Sibling fields

Degree 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 40 siblings: 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 R ${\href{/padicField/3.5.0.1}{5} }^{2}$ ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ R ${\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.5.0.1}{5} }^{2}$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ R ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.5.0.1}{5} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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 2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
\(11\) Copy content Toggle raw display 11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.2.1$x^{3} + 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.3.2.1$x^{3} + 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
\(29\) Copy content Toggle raw display $\Q_{29}$$x + 27$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 27$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} + 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.6.3.1$x^{6} + 2088 x^{5} + 1453339 x^{4} + 337280262 x^{3} + 45109591 x^{2} + 715298970 x + 9129666150$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(113\) Copy content Toggle raw display 113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.3.2.1$x^{3} + 113$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
113.3.2.1$x^{3} + 113$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
3.6180196.12t33.a.a$3$ $ 2^{2} \cdot 11^{2} \cdot 113^{2}$ 5.5.6180196.1 $A_5$ (as 5T4) $1$ $3$
3.6180196.12t33.a.b$3$ $ 2^{2} \cdot 11^{2} \cdot 113^{2}$ 5.5.6180196.1 $A_5$ (as 5T4) $1$ $3$
* 4.6180196.5t4.a.a$4$ $ 2^{2} \cdot 11^{2} \cdot 113^{2}$ 5.5.6180196.1 $A_5$ (as 5T4) $1$ $4$
* 5.437...076.10t34.a.a$5$ $ 2^{2} \cdot 11^{2} \cdot 29^{4} \cdot 113^{2}$ 10.10.27014472322230266896.1 $C_2^4 : A_5$ (as 10T34) $1$ $5$
5.381...416.6t12.a.a$5$ $ 2^{4} \cdot 11^{4} \cdot 113^{4}$ 5.5.6180196.1 $A_5$ (as 5T4) $1$ $5$
5.270...896.30t217.a.a$5$ $ 2^{4} \cdot 11^{4} \cdot 29^{4} \cdot 113^{4}$ 10.10.27014472322230266896.1 $C_2^4 : A_5$ (as 10T34) $0$ $5$
5.270...896.30t217.a.b$5$ $ 2^{4} \cdot 11^{4} \cdot 29^{4} \cdot 113^{4}$ 10.10.27014472322230266896.1 $C_2^4 : A_5$ (as 10T34) $0$ $5$
10.166...616.16t1081.a.a$10$ $ 2^{6} \cdot 11^{6} \cdot 29^{4} \cdot 113^{6}$ 10.10.27014472322230266896.1 $C_2^4 : A_5$ (as 10T34) $1$ $10$
10.166...616.20t177.a.a$10$ $ 2^{6} \cdot 11^{6} \cdot 29^{4} \cdot 113^{6}$ 10.10.27014472322230266896.1 $C_2^4 : A_5$ (as 10T34) $1$ $10$
15.318...016.30t214.a.a$15$ $ 2^{10} \cdot 11^{10} \cdot 29^{12} \cdot 113^{10}$ 10.10.27014472322230266896.1 $C_2^4 : A_5$ (as 10T34) $1$ $15$
20.172...376.40t942.a.a$20$ $ 2^{14} \cdot 11^{14} \cdot 29^{8} \cdot 113^{14}$ 10.10.27014472322230266896.1 $C_2^4 : A_5$ (as 10T34) $1$ $20$

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 *.