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Properties

Label 10.0.296...875.1
Degree $10$
Signature $[0, 5]$
Discriminant $-2.961\times 10^{26}$
Root discriminant \(443.75\)
Ramified primes $3,5,41$
Class number $75$ (GRH)
Class group [5, 15] (GRH)
Galois group $A_5\times C_2$ (as 10T11)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^10 + 205*x^8 + 36285*x^6 - 767889*x^5 + 5715400*x^4 - 21180600*x^3 + 18877630*x^2 + 65659860*x + 43616907)
 
Copy content gp:K = bnfinit(y^10 + 205*y^8 + 36285*y^6 - 767889*y^5 + 5715400*y^4 - 21180600*y^3 + 18877630*y^2 + 65659860*y + 43616907, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 + 205*x^8 + 36285*x^6 - 767889*x^5 + 5715400*x^4 - 21180600*x^3 + 18877630*x^2 + 65659860*x + 43616907);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 + 205*x^8 + 36285*x^6 - 767889*x^5 + 5715400*x^4 - 21180600*x^3 + 18877630*x^2 + 65659860*x + 43616907)
 

\( x^{10} + 205 x^{8} + 36285 x^{6} - 767889 x^{5} + 5715400 x^{4} - 21180600 x^{3} + 18877630 x^{2} + \cdots + 43616907 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $10$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[0, 5]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-296071904094910125732421875\) \(\medspace = -\,3^{5}\cdot 5^{16}\cdot 41^{8}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(443.75\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}5^{8/5}41^{4/5}\approx 443.7513830832364$
Ramified primes:   \(3\), \(5\), \(41\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{-3}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{123}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{123}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{123}a^{7}-\frac{1}{3}a$, $\frac{1}{6053691}a^{8}+\frac{12637}{6053691}a^{7}-\frac{479}{288271}a^{6}-\frac{998}{2017897}a^{5}-\frac{10358}{49217}a^{4}+\frac{6050}{49217}a^{3}+\frac{42608}{147651}a^{2}+\frac{6122}{147651}a-\frac{14675}{49217}$, $\frac{1}{19\cdots 81}a^{9}-\frac{62542869266}{63\cdots 51}a^{8}-\frac{12\cdots 96}{65\cdots 27}a^{7}+\frac{21\cdots 89}{63\cdots 51}a^{6}-\frac{34\cdots 85}{93\cdots 61}a^{5}-\frac{19\cdots 65}{47\cdots 41}a^{4}-\frac{19\cdots 84}{47\cdots 41}a^{3}-\frac{37\cdots 25}{47\cdots 41}a^{2}+\frac{67\cdots 86}{15\cdots 47}a+\frac{10\cdots 01}{51\cdots 37}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  $C_{5}\times C_{15}$, which has order $75$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{5}\times C_{15}$, which has order $75$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $4$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( \frac{24633250}{39911933654592393} a^{9} + \frac{180234785}{1287481730793303} a^{8} + \frac{27854147230}{39911933654592393} a^{7} + \frac{13571419985}{429160576931101} a^{6} + \frac{2075753560613}{13303977884864131} a^{5} + \frac{1723268669625}{324487265484491} a^{4} - \frac{77006311085860}{973461796453473} a^{3} + \frac{464239961384440}{973461796453473} a^{2} - \frac{1162768760687215}{973461796453473} a - \frac{5850603705278}{10467331144661} \)  (order $6$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{546264208245130}{63\cdots 51}a^{9}+\frac{24\cdots 86}{63\cdots 51}a^{8}+\frac{12\cdots 51}{63\cdots 51}a^{7}+\frac{21\cdots 35}{23\cdots 53}a^{6}+\frac{56\cdots 85}{15\cdots 11}a^{5}-\frac{36\cdots 40}{73\cdots 91}a^{4}+\frac{21\cdots 73}{73\cdots 91}a^{3}-\frac{10\cdots 43}{15\cdots 11}a^{2}-\frac{77\cdots 10}{51\cdots 37}a-\frac{45\cdots 34}{51\cdots 37}$, $\frac{97\cdots 38}{93\cdots 61}a^{9}+\frac{27\cdots 02}{63\cdots 51}a^{8}+\frac{44\cdots 06}{19\cdots 81}a^{7}+\frac{83\cdots 13}{90\cdots 93}a^{6}+\frac{65\cdots 83}{15\cdots 47}a^{5}-\frac{30\cdots 80}{47\cdots 41}a^{4}+\frac{51\cdots 64}{15\cdots 47}a^{3}-\frac{39\cdots 84}{47\cdots 41}a^{2}-\frac{85\cdots 15}{47\cdots 41}a-\frac{59\cdots 85}{57\cdots 33}$, $\frac{32\cdots 22}{21\cdots 17}a^{9}+\frac{11\cdots 10}{63\cdots 51}a^{8}+\frac{27\cdots 58}{63\cdots 51}a^{7}-\frac{26\cdots 63}{23\cdots 53}a^{6}+\frac{13\cdots 30}{15\cdots 11}a^{5}-\frac{26\cdots 66}{73\cdots 91}a^{4}+\frac{10\cdots 85}{22\cdots 73}a^{3}+\frac{10\cdots 86}{15\cdots 11}a^{2}-\frac{35\cdots 30}{51\cdots 37}a-\frac{22\cdots 44}{51\cdots 37}$, $\frac{52\cdots 88}{65\cdots 27}a^{9}-\frac{14\cdots 64}{30\cdots 31}a^{8}-\frac{25\cdots 73}{65\cdots 27}a^{7}-\frac{77\cdots 42}{21\cdots 17}a^{6}+\frac{19\cdots 52}{19\cdots 81}a^{5}-\frac{14\cdots 16}{15\cdots 47}a^{4}+\frac{18\cdots 83}{47\cdots 41}a^{3}-\frac{60\cdots 47}{15\cdots 47}a^{2}-\frac{59\cdots 36}{47\cdots 41}a-\frac{41\cdots 81}{51\cdots 37}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 102270212.80456363 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 102270212.80456363 \cdot 75}{6\cdot\sqrt{296071904094910125732421875}}\cr\approx \mathstrut & 0.727545093246026 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^10 + 205*x^8 + 36285*x^6 - 767889*x^5 + 5715400*x^4 - 21180600*x^3 + 18877630*x^2 + 65659860*x + 43616907) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^10 + 205*x^8 + 36285*x^6 - 767889*x^5 + 5715400*x^4 - 21180600*x^3 + 18877630*x^2 + 65659860*x + 43616907, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 + 205*x^8 + 36285*x^6 - 767889*x^5 + 5715400*x^4 - 21180600*x^3 + 18877630*x^2 + 65659860*x + 43616907); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 + 205*x^8 + 36285*x^6 - 767889*x^5 + 5715400*x^4 - 21180600*x^3 + 18877630*x^2 + 65659860*x + 43616907); 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

$C_2\times A_5$ (as 10T11):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A non-solvable group of order 120
The 10 conjugacy class representatives for $A_5\times C_2$
Character table for $A_5\times C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.1.9934316015625.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 12 siblings: 12.4.98690634698303375244140625.1, deg 12
Degree 20 siblings: deg 20, deg 20
Degree 24 sibling: deg 24
Degree 30 siblings: deg 30, deg 30
Degree 40 sibling: deg 40
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 ${\href{/padicField/2.10.0.1}{10} }$ R R ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ ${\href{/padicField/11.10.0.1}{10} }$ ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }$ ${\href{/padicField/29.10.0.1}{10} }$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ R ${\href{/padicField/43.5.0.1}{5} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.10.0.1}{10} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 3.1.2.1a1.1$x^{2} + 3$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(5\) Copy content Toggle raw display 5.2.5.16a4.5$x^{10} + 20 x^{9} + 190 x^{8} + 1120 x^{7} + 4360 x^{6} + 11104 x^{5} + 17840 x^{4} + 17280 x^{3} + 9680 x^{2} + 2880 x + 457$$5$$2$$16$$C_{10}$$$[2]^{2}$$
\(41\) Copy content Toggle raw display 41.2.5.8a1.1$x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$$5$$2$$8$$C_{10}$$$[\ ]_{5}^{2}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*120 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*120 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
3.3151875.12t76.a.a$3$ $ 3 \cdot 5^{4} \cdot 41^{2}$ 10.0.296071904094910125732421875.1 $A_5\times C_2$ (as 10T11) $1$ $1$
3.9455625.12t33.a.a$3$ $ 3^{2} \cdot 5^{4} \cdot 41^{2}$ 5.1.9934316015625.2 $A_5$ (as 5T4) $1$ $-1$
3.9455625.12t33.a.b$3$ $ 3^{2} \cdot 5^{4} \cdot 41^{2}$ 5.1.9934316015625.2 $A_5$ (as 5T4) $1$ $-1$
3.3151875.12t76.a.b$3$ $ 3 \cdot 5^{4} \cdot 41^{2}$ 10.0.296071904094910125732421875.1 $A_5\times C_2$ (as 10T11) $1$ $1$
*120 4.993...625.5t4.a.a$4$ $ 3^{2} \cdot 5^{8} \cdot 41^{4}$ 5.1.9934316015625.2 $A_5$ (as 5T4) $1$ $0$
*120 4.993...625.10t11.a.a$4$ $ 3^{2} \cdot 5^{8} \cdot 41^{4}$ 10.0.296071904094910125732421875.1 $A_5\times C_2$ (as 10T11) $1$ $0$
5.993...625.6t12.a.a$5$ $ 3^{2} \cdot 5^{8} \cdot 41^{4}$ 5.1.9934316015625.2 $A_5$ (as 5T4) $1$ $1$
5.298...875.12t75.a.a$5$ $ 3^{3} \cdot 5^{8} \cdot 41^{4}$ 10.0.296071904094910125732421875.1 $A_5\times C_2$ (as 10T11) $1$ $-1$

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 *.

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)