Properties

Label 24.24.510...000.1
Degree $24$
Signature $[24, 0]$
Discriminant $5.107\times 10^{36}$
Root discriminant \(33.85\)
Ramified primes $2,5,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 23*x^22 + 230*x^20 - 1311*x^18 + 4692*x^16 - 10949*x^14 + 16757*x^12 - 16511*x^10 + 10032*x^8 - 3498*x^6 + 628*x^4 - 48*x^2 + 1)
 
gp: K = bnfinit(y^24 - 23*y^22 + 230*y^20 - 1311*y^18 + 4692*y^16 - 10949*y^14 + 16757*y^12 - 16511*y^10 + 10032*y^8 - 3498*y^6 + 628*y^4 - 48*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 23*x^22 + 230*x^20 - 1311*x^18 + 4692*x^16 - 10949*x^14 + 16757*x^12 - 16511*x^10 + 10032*x^8 - 3498*x^6 + 628*x^4 - 48*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 23*x^22 + 230*x^20 - 1311*x^18 + 4692*x^16 - 10949*x^14 + 16757*x^12 - 16511*x^10 + 10032*x^8 - 3498*x^6 + 628*x^4 - 48*x^2 + 1)
 

\( x^{24} - 23 x^{22} + 230 x^{20} - 1311 x^{18} + 4692 x^{16} - 10949 x^{14} + 16757 x^{12} - 16511 x^{10} + 10032 x^{8} - 3498 x^{6} + 628 x^{4} - 48 x^{2} + 1 \) 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:   \(5106705043047168064000000000000000000\) \(\medspace = 2^{24}\cdot 5^{18}\cdot 7^{20}\) 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:  \(33.85\)
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:  $2\cdot 5^{3/4}7^{5/6}\approx 33.8458843070916$
Ramified primes:   \(2\), \(5\), \(7\) 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:  \(140=2^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{140}(1,·)$, $\chi_{140}(131,·)$, $\chi_{140}(97,·)$, $\chi_{140}(9,·)$, $\chi_{140}(139,·)$, $\chi_{140}(13,·)$, $\chi_{140}(17,·)$, $\chi_{140}(19,·)$, $\chi_{140}(23,·)$, $\chi_{140}(29,·)$, $\chi_{140}(31,·)$, $\chi_{140}(107,·)$, $\chi_{140}(33,·)$, $\chi_{140}(67,·)$, $\chi_{140}(123,·)$, $\chi_{140}(81,·)$, $\chi_{140}(43,·)$, $\chi_{140}(109,·)$, $\chi_{140}(111,·)$, $\chi_{140}(117,·)$, $\chi_{140}(73,·)$, $\chi_{140}(121,·)$, $\chi_{140}(59,·)$, $\chi_{140}(127,·)$$\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:   $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-3$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+3$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-2$, $2a^{23}-46a^{21}+459a^{19}-2602a^{17}+9215a^{15}-21113a^{13}+31330a^{11}-29304a^{9}+16268a^{7}-4811a^{5}+626a^{3}-23a$, $a$, $2a^{23}-46a^{21}+459a^{19}-2602a^{17}+9214a^{15}-21099a^{13}+31253a^{11}-29094a^{9}+15974a^{7}-4615a^{5}+577a^{3}-20a$, $a^{3}-3a$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a$, $a^{19}-19a^{17}+152a^{15}-665a^{13}+1729a^{11}-2717a^{9}+2508a^{7}-1254a^{5}+285a^{3}-19a$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a$, $a^{23}-23a^{21}+229a^{19}-1292a^{17}+4540a^{15}-10283a^{13}+15015a^{11}-13729a^{9}+7368a^{7}-2061a^{5}+246a^{3}-8a$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+a^{7}+294a^{6}-7a^{5}-196a^{4}+14a^{3}+49a^{2}-7a-2$, $2a^{23}-45a^{21}+439a^{19}-a^{18}-2433a^{17}+18a^{16}+8431a^{15}-135a^{14}-18943a^{13}+546a^{12}+27689a^{11}-1288a^{10}-25718a^{9}+1792a^{8}+14377a^{7}-1421a^{6}-4377a^{5}+590a^{4}+601a^{3}-106a^{2}-21a+4$, $a^{6}-6a^{4}+9a^{2}+a-2$, $2a^{23}+a^{22}-45a^{21}-22a^{20}+439a^{19}+210a^{18}-2432a^{17}-1140a^{16}+8414a^{15}+3874a^{14}-18824a^{13}-8541a^{12}+27247a^{11}+12232a^{10}-24783a^{9}-11055a^{8}+13255a^{7}+5896a^{6}-3663a^{5}-1629a^{4}+398a^{3}+181a^{2}-7a-5$, $a^{20}+a^{19}-20a^{18}-19a^{17}+169a^{16}+152a^{15}-784a^{14}-665a^{13}+2170a^{12}+1729a^{11}-3641a^{10}-2717a^{9}+3586a^{8}+2508a^{7}-1891a^{6}-1254a^{5}+434a^{4}+285a^{3}-24a^{2}-19a-1$, $a^{23}-23a^{21}-a^{20}+229a^{19}+19a^{18}-1292a^{17}-151a^{16}+4540a^{15}+650a^{14}-10283a^{13}-1638a^{12}+15016a^{11}+2431a^{10}-13740a^{9}-2014a^{8}+7411a^{7}+799a^{6}-2131a^{5}-90a^{4}+287a^{3}-8a^{2}-12a+1$, $2a^{23}-46a^{21}-a^{20}+459a^{19}+20a^{18}-2602a^{17}-170a^{16}+9215a^{15}+800a^{14}-21113a^{13}-2275a^{12}+31330a^{11}+4004a^{10}-29304a^{9}-4290a^{8}+16268a^{7}+2640a^{6}-4811a^{5}-825a^{4}+626a^{3}+100a^{2}-23a-2$, $2a^{23}-46a^{21}+459a^{19}-2602a^{17}+9215a^{15}-a^{14}-21113a^{13}+14a^{12}+31330a^{11}-77a^{10}-29304a^{9}+210a^{8}+16268a^{7}-294a^{6}-4811a^{5}+196a^{4}+626a^{3}-49a^{2}-23a+3$, $a^{22}-22a^{20}-a^{19}+210a^{18}+19a^{17}-1140a^{16}-152a^{15}+3874a^{14}+666a^{13}-8541a^{12}-1742a^{11}+12233a^{10}+2782a^{9}-11065a^{8}-2664a^{7}+5931a^{6}+1436a^{5}-1679a^{4}-376a^{3}+206a^{2}+32a-8$, $a^{22}-23a^{20}+229a^{18}-a^{17}-1292a^{16}+17a^{15}+4540a^{14}-119a^{13}-10283a^{12}+442a^{11}+15015a^{10}-935a^{9}-13729a^{8}+1122a^{7}+7368a^{6}-714a^{5}-2061a^{4}+204a^{3}+246a^{2}-17a-9$, $2a^{23}-46a^{21}+460a^{19}-2621a^{17}+9366a^{15}-21764a^{13}+32982a^{11}-31811a^{9}+18482a^{7}-5869a^{5}+862a^{3}-39a+1$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4005a^{10}-a^{9}+4300a^{8}+9a^{7}-2675a^{6}-27a^{5}+875a^{4}+30a^{3}-125a^{2}-9a+5$ 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:  \( 33400075541.82508 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{24}\cdot(2\pi)^{0}\cdot 33400075541.82508 \cdot 1}{2\cdot\sqrt{5106705043047168064000000000000000000}}\cr\approx \mathstrut & 0.123984376455386 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - 23*x^22 + 230*x^20 - 1311*x^18 + 4692*x^16 - 10949*x^14 + 16757*x^12 - 16511*x^10 + 10032*x^8 - 3498*x^6 + 628*x^4 - 48*x^2 + 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 - 23*x^22 + 230*x^20 - 1311*x^18 + 4692*x^16 - 10949*x^14 + 16757*x^12 - 16511*x^10 + 10032*x^8 - 3498*x^6 + 628*x^4 - 48*x^2 + 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 - 23*x^22 + 230*x^20 - 1311*x^18 + 4692*x^16 - 10949*x^14 + 16757*x^12 - 16511*x^10 + 10032*x^8 - 3498*x^6 + 628*x^4 - 48*x^2 + 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 - 23*x^22 + 230*x^20 - 1311*x^18 + 4692*x^16 - 10949*x^14 + 16757*x^12 - 16511*x^10 + 10032*x^8 - 3498*x^6 + 628*x^4 - 48*x^2 + 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 $C_2\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{5}, \sqrt{7})\), 4.4.6125.1, \(\Q(\zeta_{20})^+\), 6.6.134456000.1, 6.6.300125.1, \(\Q(\zeta_{28})^+\), 8.8.9604000000.1, 12.12.18078415936000000.1, \(\Q(\zeta_{35})^+\), 12.12.46118408000000000.1

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 R ${\href{/padicField/3.12.0.1}{12} }^{2}$ R R ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{6}$ ${\href{/padicField/17.12.0.1}{12} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{8}$ ${\href{/padicField/23.12.0.1}{12} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{12}$ ${\href{/padicField/43.4.0.1}{4} }^{6}$ ${\href{/padicField/47.12.0.1}{12} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{8}$

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 Deg $24$$2$$12$$24$
\(5\) Copy content Toggle raw display Deg $24$$4$$6$$18$
\(7\) Copy content Toggle raw display Deg $24$$6$$4$$20$