Properties

Label 24.4.918...125.1
Degree $24$
Signature $(4, 10)$
Discriminant $9.181\times 10^{58}$
Root discriminant \(286.28\)
Ramified primes $5,89$
Class number not computed
Class group not computed
Galois group $\GL(2,5)$ (as 24T1353)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^23 + 6*x^22 - 449*x^21 + 4273*x^20 + 5874*x^19 + 40584*x^18 - 1486656*x^17 + 320934*x^16 + 31758582*x^15 + 203615891*x^14 - 1041728624*x^13 - 8957494979*x^12 + 15522090746*x^11 + 187723100658*x^10 - 120260671411*x^9 - 2012872155141*x^8 + 555294801359*x^7 + 10666291776869*x^6 + 2324146207637*x^5 - 37689124873554*x^4 - 13595215964929*x^3 + 96703661945176*x^2 - 15091490947869*x - 91903199858837)
 
Copy content gp:K = bnfinit(y^24 - 4*y^23 + 6*y^22 - 449*y^21 + 4273*y^20 + 5874*y^19 + 40584*y^18 - 1486656*y^17 + 320934*y^16 + 31758582*y^15 + 203615891*y^14 - 1041728624*y^13 - 8957494979*y^12 + 15522090746*y^11 + 187723100658*y^10 - 120260671411*y^9 - 2012872155141*y^8 + 555294801359*y^7 + 10666291776869*y^6 + 2324146207637*y^5 - 37689124873554*y^4 - 13595215964929*y^3 + 96703661945176*y^2 - 15091490947869*y - 91903199858837, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 4*x^23 + 6*x^22 - 449*x^21 + 4273*x^20 + 5874*x^19 + 40584*x^18 - 1486656*x^17 + 320934*x^16 + 31758582*x^15 + 203615891*x^14 - 1041728624*x^13 - 8957494979*x^12 + 15522090746*x^11 + 187723100658*x^10 - 120260671411*x^9 - 2012872155141*x^8 + 555294801359*x^7 + 10666291776869*x^6 + 2324146207637*x^5 - 37689124873554*x^4 - 13595215964929*x^3 + 96703661945176*x^2 - 15091490947869*x - 91903199858837);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 - 4*x^23 + 6*x^22 - 449*x^21 + 4273*x^20 + 5874*x^19 + 40584*x^18 - 1486656*x^17 + 320934*x^16 + 31758582*x^15 + 203615891*x^14 - 1041728624*x^13 - 8957494979*x^12 + 15522090746*x^11 + 187723100658*x^10 - 120260671411*x^9 - 2012872155141*x^8 + 555294801359*x^7 + 10666291776869*x^6 + 2324146207637*x^5 - 37689124873554*x^4 - 13595215964929*x^3 + 96703661945176*x^2 - 15091490947869*x - 91903199858837)
 

\( x^{24} - 4 x^{23} + 6 x^{22} - 449 x^{21} + 4273 x^{20} + 5874 x^{19} + 40584 x^{18} + \cdots - 91903199858837 \) 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:  $24$
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:  $(4, 10)$
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:   \(91810054652229848026521481130092527303208363056182861328125\) \(\medspace = 5^{23}\cdot 89^{22}\) 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:  \(286.28\)
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:  $5^{23/20}89^{19/20}\approx 452.62236684313837$
Ramified primes:   \(5\), \(89\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_4$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{9}-\frac{1}{5}a^{8}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{5}a^{14}-\frac{2}{5}a^{10}+\frac{2}{5}a^{9}-\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{15}-\frac{2}{5}a^{10}+\frac{2}{5}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{16}-\frac{2}{5}a^{10}+\frac{1}{5}a^{8}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{3}+\frac{1}{5}$, $\frac{1}{5}a^{17}+\frac{1}{5}a^{10}-\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}$, $\frac{1}{25}a^{18}+\frac{1}{25}a^{16}-\frac{2}{25}a^{14}-\frac{2}{25}a^{12}-\frac{7}{25}a^{10}-\frac{9}{25}a^{9}+\frac{2}{25}a^{8}+\frac{11}{25}a^{7}-\frac{8}{25}a^{6}+\frac{1}{25}a^{5}-\frac{7}{25}a^{4}+\frac{3}{25}a^{3}-\frac{12}{25}a^{2}-\frac{7}{25}a+\frac{8}{25}$, $\frac{1}{25}a^{19}+\frac{1}{25}a^{17}-\frac{2}{25}a^{15}-\frac{2}{25}a^{13}-\frac{2}{25}a^{11}+\frac{6}{25}a^{10}-\frac{3}{25}a^{9}+\frac{6}{25}a^{8}-\frac{3}{25}a^{7}-\frac{9}{25}a^{6}-\frac{2}{25}a^{5}-\frac{2}{25}a^{4}+\frac{8}{25}a^{3}-\frac{2}{25}a^{2}-\frac{7}{25}a-\frac{2}{5}$, $\frac{1}{25}a^{20}+\frac{2}{25}a^{16}+\frac{1}{25}a^{11}+\frac{4}{25}a^{10}-\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{11}{25}a^{6}+\frac{12}{25}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{12}{25}a+\frac{7}{25}$, $\frac{1}{25}a^{21}+\frac{2}{25}a^{17}+\frac{1}{25}a^{12}-\frac{1}{25}a^{11}+\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{6}{25}a^{7}-\frac{3}{25}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{7}{25}a^{2}-\frac{3}{25}a+\frac{2}{5}$, $\frac{1}{25}a^{22}-\frac{2}{25}a^{16}-\frac{1}{25}a^{14}+\frac{1}{25}a^{13}-\frac{2}{25}a^{12}-\frac{6}{25}a^{10}+\frac{8}{25}a^{9}-\frac{8}{25}a^{8}+\frac{1}{25}a^{6}-\frac{12}{25}a^{5}+\frac{9}{25}a^{4}-\frac{9}{25}a^{3}+\frac{1}{25}a^{2}-\frac{1}{25}a-\frac{6}{25}$, $\frac{1}{34\cdots 25}a^{23}-\frac{96\cdots 71}{69\cdots 65}a^{22}-\frac{24\cdots 31}{34\cdots 25}a^{21}-\frac{17\cdots 12}{34\cdots 25}a^{20}-\frac{55\cdots 44}{69\cdots 65}a^{19}-\frac{17\cdots 86}{34\cdots 25}a^{18}-\frac{19\cdots 29}{34\cdots 25}a^{17}+\frac{69\cdots 92}{69\cdots 65}a^{16}+\frac{91\cdots 04}{34\cdots 25}a^{15}+\frac{66\cdots 08}{34\cdots 25}a^{14}+\frac{32\cdots 13}{34\cdots 25}a^{13}+\frac{28\cdots 46}{34\cdots 25}a^{12}-\frac{15\cdots 97}{34\cdots 25}a^{11}-\frac{94\cdots 03}{34\cdots 25}a^{10}-\frac{16\cdots 94}{34\cdots 25}a^{9}-\frac{11\cdots 87}{34\cdots 25}a^{8}+\frac{16\cdots 84}{34\cdots 25}a^{7}-\frac{14\cdots 03}{34\cdots 25}a^{6}-\frac{83\cdots 76}{34\cdots 25}a^{5}-\frac{17\cdots 22}{34\cdots 25}a^{4}-\frac{14\cdots 46}{74\cdots 75}a^{3}-\frac{16\cdots 11}{34\cdots 25}a^{2}-\frac{26\cdots 93}{69\cdots 65}a+\frac{65\cdots 18}{34\cdots 25}$ 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:  not computed
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:  not computed
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:  $13$
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:   \( -1 \)  (order $2$) 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:  not computed
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:  not computed
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot R \cdot h}{2\cdot\sqrt{91810054652229848026521481130092527303208363056182861328125}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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^24 - 4*x^23 + 6*x^22 - 449*x^21 + 4273*x^20 + 5874*x^19 + 40584*x^18 - 1486656*x^17 + 320934*x^16 + 31758582*x^15 + 203615891*x^14 - 1041728624*x^13 - 8957494979*x^12 + 15522090746*x^11 + 187723100658*x^10 - 120260671411*x^9 - 2012872155141*x^8 + 555294801359*x^7 + 10666291776869*x^6 + 2324146207637*x^5 - 37689124873554*x^4 - 13595215964929*x^3 + 96703661945176*x^2 - 15091490947869*x - 91903199858837) 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^24 - 4*x^23 + 6*x^22 - 449*x^21 + 4273*x^20 + 5874*x^19 + 40584*x^18 - 1486656*x^17 + 320934*x^16 + 31758582*x^15 + 203615891*x^14 - 1041728624*x^13 - 8957494979*x^12 + 15522090746*x^11 + 187723100658*x^10 - 120260671411*x^9 - 2012872155141*x^8 + 555294801359*x^7 + 10666291776869*x^6 + 2324146207637*x^5 - 37689124873554*x^4 - 13595215964929*x^3 + 96703661945176*x^2 - 15091490947869*x - 91903199858837, 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^24 - 4*x^23 + 6*x^22 - 449*x^21 + 4273*x^20 + 5874*x^19 + 40584*x^18 - 1486656*x^17 + 320934*x^16 + 31758582*x^15 + 203615891*x^14 - 1041728624*x^13 - 8957494979*x^12 + 15522090746*x^11 + 187723100658*x^10 - 120260671411*x^9 - 2012872155141*x^8 + 555294801359*x^7 + 10666291776869*x^6 + 2324146207637*x^5 - 37689124873554*x^4 - 13595215964929*x^3 + 96703661945176*x^2 - 15091490947869*x - 91903199858837); 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^24 - 4*x^23 + 6*x^22 - 449*x^21 + 4273*x^20 + 5874*x^19 + 40584*x^18 - 1486656*x^17 + 320934*x^16 + 31758582*x^15 + 203615891*x^14 - 1041728624*x^13 - 8957494979*x^12 + 15522090746*x^11 + 187723100658*x^10 - 120260671411*x^9 - 2012872155141*x^8 + 555294801359*x^7 + 10666291776869*x^6 + 2324146207637*x^5 - 37689124873554*x^4 - 13595215964929*x^3 + 96703661945176*x^2 - 15091490947869*x - 91903199858837); 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

$\GL(2,5)$ (as 24T1353):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
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 480
The 24 conjugacy class representatives for $\GL(2,5)$
Character table for $\GL(2,5)$

Intermediate fields

6.2.196069503125.2, 12.4.1522544918455380058642578125.2

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 24 siblings: 24.4.2295251366305746200663037028252313182580209076404571533203125.4, 24.4.2295251366305746200663037028252313182580209076404571533203125.6
Arithmetically equivalent sibling: 24.4.91810054652229848026521481130092527303208363056182861328125.3
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 $24$ $24$ R ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{4}$ $24$ $24$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ $24$ ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.12.0.1}{12} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{3}$ ${\href{/padicField/41.12.0.1}{12} }^{2}$ $24$ ${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{6}$

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
\(5\) Copy content Toggle raw display 5.4.1.0a1.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
5.1.20.23a3.3$x^{20} + 20 x^{4} + 15$$20$$1$$23$20T20$not computed$
\(89\) Copy content Toggle raw display 89.1.4.3a1.1$x^{4} + 89$$4$$1$$3$$C_4$$$[\ ]_{4}$$
89.1.20.19a1.3$x^{20} + 801$$20$$1$$19$20T6$$[\ ]_{20}^{2}$$

Spectrum of ring of integers

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