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Properties

Label 24.24.122...625.1
Degree $24$
Signature $[24, 0]$
Discriminant $1.225\times 10^{37}$
Root discriminant \(35.10\)
Ramified primes $5,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 24*x^22 + 23*x^21 + 252*x^20 - 229*x^19 - 1521*x^18 + 1292*x^17 + 5832*x^16 - 4540*x^15 - 14822*x^14 + 10282*x^13 + 25284*x^12 - 15001*x^11 - 28667*x^10 + 13653*x^9 + 20886*x^8 - 7168*x^7 - 9126*x^6 + 1802*x^5 + 2085*x^4 - 101*x^3 - 180*x^2 - 12*x + 1)
 
gp: K = bnfinit(y^24 - y^23 - 24*y^22 + 23*y^21 + 252*y^20 - 229*y^19 - 1521*y^18 + 1292*y^17 + 5832*y^16 - 4540*y^15 - 14822*y^14 + 10282*y^13 + 25284*y^12 - 15001*y^11 - 28667*y^10 + 13653*y^9 + 20886*y^8 - 7168*y^7 - 9126*y^6 + 1802*y^5 + 2085*y^4 - 101*y^3 - 180*y^2 - 12*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 - 24*x^22 + 23*x^21 + 252*x^20 - 229*x^19 - 1521*x^18 + 1292*x^17 + 5832*x^16 - 4540*x^15 - 14822*x^14 + 10282*x^13 + 25284*x^12 - 15001*x^11 - 28667*x^10 + 13653*x^9 + 20886*x^8 - 7168*x^7 - 9126*x^6 + 1802*x^5 + 2085*x^4 - 101*x^3 - 180*x^2 - 12*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 - 24*x^22 + 23*x^21 + 252*x^20 - 229*x^19 - 1521*x^18 + 1292*x^17 + 5832*x^16 - 4540*x^15 - 14822*x^14 + 10282*x^13 + 25284*x^12 - 15001*x^11 - 28667*x^10 + 13653*x^9 + 20886*x^8 - 7168*x^7 - 9126*x^6 + 1802*x^5 + 2085*x^4 - 101*x^3 - 180*x^2 - 12*x + 1)
 

\( x^{24} - x^{23} - 24 x^{22} + 23 x^{21} + 252 x^{20} - 229 x^{19} - 1521 x^{18} + 1292 x^{17} + 5832 x^{16} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $24$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[24, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(12252192985362453861836887359619140625\) \(\medspace = 5^{18}\cdot 13^{22}\) 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:  \(35.10\)
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:  $5^{3/4}13^{11/12}\approx 35.10283905640381$
Ramified primes:   \(5\), \(13\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $24$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(65=5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{65}(64,·)$, $\chi_{65}(1,·)$, $\chi_{65}(2,·)$, $\chi_{65}(4,·)$, $\chi_{65}(7,·)$, $\chi_{65}(8,·)$, $\chi_{65}(9,·)$, $\chi_{65}(14,·)$, $\chi_{65}(16,·)$, $\chi_{65}(18,·)$, $\chi_{65}(28,·)$, $\chi_{65}(29,·)$, $\chi_{65}(32,·)$, $\chi_{65}(33,·)$, $\chi_{65}(36,·)$, $\chi_{65}(37,·)$, $\chi_{65}(47,·)$, $\chi_{65}(49,·)$, $\chi_{65}(51,·)$, $\chi_{65}(56,·)$, $\chi_{65}(57,·)$, $\chi_{65}(58,·)$, $\chi_{65}(61,·)$, $\chi_{65}(63,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

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:  $23$
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:   $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+2$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-2$, $a^{22}-23a^{20}+229a^{18}+a^{17}-1292a^{16}-18a^{15}+4540a^{14}+134a^{13}-10283a^{12}-532a^{11}+15014a^{10}+1211a^{9}-13718a^{8}-1581a^{7}+7324a^{6}+1118a^{5}-1984a^{4}-369a^{3}+192a^{2}+36a-1$, $a^{22}-23a^{20}+229a^{18}+a^{17}-1292a^{16}-18a^{15}+4540a^{14}+134a^{13}-10283a^{12}-532a^{11}+15015a^{10}+1211a^{9}-13728a^{8}-1581a^{7}+7359a^{6}+1118a^{5}-2034a^{4}-369a^{3}+217a^{2}+36a-3$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-377a^{5}+135a^{3}-10a$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a^{22}-23a^{20}+229a^{18}+a^{17}-1292a^{16}-18a^{15}+4541a^{14}+134a^{13}-10296a^{12}-532a^{11}+15079a^{10}+1211a^{9}-13874a^{8}-1581a^{7}+7506a^{6}+1118a^{5}-2075a^{4}-369a^{3}+205a^{2}+37a-1$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+a^{16}+4692a^{15}-16a^{14}-10948a^{13}+104a^{12}+16744a^{11}-351a^{10}-16445a^{9}+650a^{8}+9867a^{7}-637a^{6}-3289a^{5}+286a^{4}+507a^{3}-39a^{2}-26a$, $a^{3}-3a$, $a^{13}+a^{12}-13a^{11}-12a^{10}+65a^{9}+54a^{8}-156a^{7}-112a^{6}+182a^{5}+105a^{4}-91a^{3}-36a^{2}+13a+3$, $2a^{23}-46a^{21}+459a^{19}-2604a^{17}+9249a^{15}-a^{14}-21348a^{13}+15a^{12}+32175a^{11}-87a^{10}-30979a^{9}+244a^{8}+18044a^{7}-338a^{6}-5694a^{5}+211a^{4}+766a^{3}-44a^{2}-23a$, $a^{22}-23a^{20}+229a^{18}+a^{17}-1292a^{16}-18a^{15}+4540a^{14}+133a^{13}-10283a^{12}-519a^{11}+15014a^{10}+1146a^{9}-13718a^{8}-1425a^{7}+7324a^{6}+936a^{5}-1984a^{4}-278a^{3}+192a^{2}+23a-1$, $a^{15}-14a^{13}+77a^{11}-210a^{9}+294a^{7}-196a^{5}+49a^{3}-2a+1$, $a^{23}-23a^{21}+229a^{19}+a^{18}-1292a^{17}-18a^{16}+4540a^{15}+134a^{14}-10283a^{13}-532a^{12}+15014a^{11}+1211a^{10}-13718a^{9}-1581a^{8}+7324a^{7}+1118a^{6}-1984a^{5}-369a^{4}+192a^{3}+36a^{2}-a+1$, $a^{13}-13a^{11}+65a^{9}-155a^{7}+175a^{5}-77a^{3}+6a+1$, $a$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}-a^{10}+7007a^{9}+10a^{8}-5148a^{7}-35a^{6}+2079a^{5}+50a^{4}-385a^{3}-25a^{2}+21a+2$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+2$, $a^{23}-24a^{21}+a^{20}+251a^{19}-20a^{18}-1500a^{17}+170a^{16}+5645a^{15}-801a^{14}-13903a^{13}+2288a^{12}+22567a^{11}-4068a^{10}-23728a^{9}+4436a^{8}+15474a^{7}-2787a^{6}-5773a^{5}+865a^{4}+1062a^{3}-84a^{2}-72a-1$ 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:  \( 58142934011.05266 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{24}\cdot(2\pi)^{0}\cdot 58142934011.05266 \cdot 1}{2\cdot\sqrt{12252192985362453861836887359619140625}}\cr\approx \mathstrut & 0.139341322853607 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 24*x^22 + 23*x^21 + 252*x^20 - 229*x^19 - 1521*x^18 + 1292*x^17 + 5832*x^16 - 4540*x^15 - 14822*x^14 + 10282*x^13 + 25284*x^12 - 15001*x^11 - 28667*x^10 + 13653*x^9 + 20886*x^8 - 7168*x^7 - 9126*x^6 + 1802*x^5 + 2085*x^4 - 101*x^3 - 180*x^2 - 12*x + 1)
 
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^24 - x^23 - 24*x^22 + 23*x^21 + 252*x^20 - 229*x^19 - 1521*x^18 + 1292*x^17 + 5832*x^16 - 4540*x^15 - 14822*x^14 + 10282*x^13 + 25284*x^12 - 15001*x^11 - 28667*x^10 + 13653*x^9 + 20886*x^8 - 7168*x^7 - 9126*x^6 + 1802*x^5 + 2085*x^4 - 101*x^3 - 180*x^2 - 12*x + 1, 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^24 - x^23 - 24*x^22 + 23*x^21 + 252*x^20 - 229*x^19 - 1521*x^18 + 1292*x^17 + 5832*x^16 - 4540*x^15 - 14822*x^14 + 10282*x^13 + 25284*x^12 - 15001*x^11 - 28667*x^10 + 13653*x^9 + 20886*x^8 - 7168*x^7 - 9126*x^6 + 1802*x^5 + 2085*x^4 - 101*x^3 - 180*x^2 - 12*x + 1);
 
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^24 - x^23 - 24*x^22 + 23*x^21 + 252*x^20 - 229*x^19 - 1521*x^18 + 1292*x^17 + 5832*x^16 - 4540*x^15 - 14822*x^14 + 10282*x^13 + 25284*x^12 - 15001*x^11 - 28667*x^10 + 13653*x^9 + 20886*x^8 - 7168*x^7 - 9126*x^6 + 1802*x^5 + 2085*x^4 - 101*x^3 - 180*x^2 - 12*x + 1);
 
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

$C_2\times C_{12}$ (as 24T2):

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)
 
An abelian group of order 24
The 24 conjugacy class representatives for $C_2\times C_{12}$
Character table for $C_2\times C_{12}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\sqrt{5}, \sqrt{13})\), 4.4.274625.1, 4.4.274625.2, 6.6.3570125.1, 6.6.46411625.1, \(\Q(\zeta_{13})^+\), 8.8.75418890625.1, 12.12.2154038935140625.1, 12.12.3500313269603515625.2, 12.12.3500313269603515625.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{4}$ ${\href{/padicField/3.12.0.1}{12} }^{2}$ R ${\href{/padicField/7.6.0.1}{6} }^{4}$ ${\href{/padicField/11.12.0.1}{12} }^{2}$ R ${\href{/padicField/17.12.0.1}{12} }^{2}$ ${\href{/padicField/19.12.0.1}{12} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{4}$ ${\href{/padicField/41.12.0.1}{12} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{12}$ ${\href{/padicField/53.4.0.1}{4} }^{6}$ ${\href{/padicField/59.12.0.1}{12} }^{2}$

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
\(5\) Copy content Toggle raw display 5.8.6.2$x^{8} + 10 x^{4} - 25$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 10 x^{4} - 25$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 10 x^{4} - 25$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(13\) Copy content Toggle raw display Deg $24$$12$$2$$22$