Properties

Label 24.24.161...625.1
Degree $24$
Signature $[24, 0]$
Discriminant $1.618\times 10^{35}$
Root discriminant \(29.31\)
Ramified primes $3,5,7$
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 - 23*x^22 + 23*x^21 + 230*x^20 - 229*x^19 - 1312*x^18 + 1294*x^17 + 4709*x^16 - 4573*x^15 - 11067*x^14 + 10506*x^13 + 17187*x^12 - 15810*x^11 - 17391*x^10 + 15333*x^9 + 11034*x^8 - 9189*x^7 - 4088*x^6 + 3142*x^5 + 784*x^4 - 528*x^3 - 64*x^2 + 32*x + 1)
 
gp: K = bnfinit(y^24 - y^23 - 23*y^22 + 23*y^21 + 230*y^20 - 229*y^19 - 1312*y^18 + 1294*y^17 + 4709*y^16 - 4573*y^15 - 11067*y^14 + 10506*y^13 + 17187*y^12 - 15810*y^11 - 17391*y^10 + 15333*y^9 + 11034*y^8 - 9189*y^7 - 4088*y^6 + 3142*y^5 + 784*y^4 - 528*y^3 - 64*y^2 + 32*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 - 23*x^22 + 23*x^21 + 230*x^20 - 229*x^19 - 1312*x^18 + 1294*x^17 + 4709*x^16 - 4573*x^15 - 11067*x^14 + 10506*x^13 + 17187*x^12 - 15810*x^11 - 17391*x^10 + 15333*x^9 + 11034*x^8 - 9189*x^7 - 4088*x^6 + 3142*x^5 + 784*x^4 - 528*x^3 - 64*x^2 + 32*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 - 23*x^22 + 23*x^21 + 230*x^20 - 229*x^19 - 1312*x^18 + 1294*x^17 + 4709*x^16 - 4573*x^15 - 11067*x^14 + 10506*x^13 + 17187*x^12 - 15810*x^11 - 17391*x^10 + 15333*x^9 + 11034*x^8 - 9189*x^7 - 4088*x^6 + 3142*x^5 + 784*x^4 - 528*x^3 - 64*x^2 + 32*x + 1)
 

\( x^{24} - x^{23} - 23 x^{22} + 23 x^{21} + 230 x^{20} - 229 x^{19} - 1312 x^{18} + 1294 x^{17} + 4709 x^{16} - 4573 x^{15} - 11067 x^{14} + 10506 x^{13} + 17187 x^{12} - 15810 x^{11} + \cdots + 1 \) 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:   \(161761786626698377317203521728515625\) \(\medspace = 3^{12}\cdot 5^{18}\cdot 7^{20}\) 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:  \(29.31\)
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:  $3^{1/2}5^{3/4}7^{5/6}\approx 29.3113956234904$
Ramified primes:   \(3\), \(5\), \(7\) 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:  \(105=3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(2,·)$, $\chi_{105}(4,·)$, $\chi_{105}(8,·)$, $\chi_{105}(73,·)$, $\chi_{105}(13,·)$, $\chi_{105}(79,·)$, $\chi_{105}(16,·)$, $\chi_{105}(82,·)$, $\chi_{105}(23,·)$, $\chi_{105}(89,·)$, $\chi_{105}(26,·)$, $\chi_{105}(92,·)$, $\chi_{105}(32,·)$, $\chi_{105}(97,·)$, $\chi_{105}(101,·)$, $\chi_{105}(103,·)$, $\chi_{105}(104,·)$, $\chi_{105}(41,·)$, $\chi_{105}(46,·)$, $\chi_{105}(52,·)$, $\chi_{105}(53,·)$, $\chi_{105}(59,·)$$\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:   $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a$, $a^{23}-22a^{21}+a^{20}+209a^{19}-18a^{18}-1122a^{17}+136a^{16}+3740a^{15}-561a^{14}-8007a^{13}+1377a^{12}+10999a^{11}-2057a^{10}-9383a^{9}+1835a^{8}+4598a^{7}-911a^{6}-1079a^{5}+206a^{4}+58a^{3}-7a^{2}+8a-1$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{20}-20a^{18}+170a^{16}-a^{15}-800a^{14}+15a^{13}+2275a^{12}-90a^{11}-4004a^{10}+275a^{9}+4290a^{8}-450a^{7}-2640a^{6}+378a^{5}+825a^{4}-140a^{3}-100a^{2}+15a+3$, $a^{6}-6a^{4}+9a^{2}+a-2$, $a^{23}-23a^{21}+a^{20}+231a^{19}-19a^{18}-1330a^{17}+153a^{16}+4845a^{15}-680a^{14}-11627a^{13}+1819a^{12}+18551a^{11}-2992a^{10}-19383a^{9}+2958a^{8}+12713a^{7}-1632a^{6}-4817a^{5}+424a^{4}+899a^{3}-31a^{2}-58a-1$, $a^{23}-23a^{21}+a^{20}+231a^{19}-19a^{18}-1330a^{17}+153a^{16}+4845a^{15}-680a^{14}-11627a^{13}+1819a^{12}+18551a^{11}-2992a^{10}-19383a^{9}+2958a^{8}+12713a^{7}-1633a^{6}-4817a^{5}+430a^{4}+899a^{3}-40a^{2}-59a+1$, $a^{5}-5a^{3}+5a+1$, $2a^{23}-a^{22}-46a^{21}+22a^{20}+460a^{19}-207a^{18}-2624a^{17}+1087a^{16}+9418a^{15}-3486a^{14}-22134a^{13}+7019a^{12}+34373a^{11}-8778a^{10}-34771a^{9}+6489a^{8}+22023a^{7}-2534a^{6}-8091a^{5}+391a^{4}+1493a^{3}+4a^{2}-100a-2$, $a^{23}-23a^{21}+231a^{19}+a^{18}-1331a^{17}-17a^{16}+4861a^{15}+120a^{14}-11732a^{13}-457a^{12}+18916a^{11}+1024a^{10}-20108a^{9}-1386a^{8}+13541a^{7}+1120a^{6}-5334a^{5}-506a^{4}+1050a^{3}+106a^{2}-71a-6$, $a^{23}+a^{22}-22a^{21}-21a^{20}+211a^{19}+191a^{18}-1158a^{17}-986a^{16}+4013a^{15}+3180a^{14}-9143a^{13}-6643a^{12}+13831a^{11}+9008a^{10}-13719a^{9}-7714a^{8}+8616a^{7}+3904a^{6}-3209a^{5}-1016a^{4}+624a^{3}+96a^{2}-45a$, $a^{23}-23a^{21}+a^{20}+231a^{19}-19a^{18}-1329a^{17}+153a^{16}+4828a^{15}-680a^{14}-11508a^{13}+1820a^{12}+18109a^{11}-3004a^{10}-18448a^{9}+3012a^{8}+11591a^{7}-1745a^{6}-4103a^{5}+535a^{4}+695a^{3}-77a^{2}-42a+5$, $a^{23}+a^{22}-22a^{21}-21a^{20}+210a^{19}+190a^{18}-1140a^{17}-969a^{16}+3876a^{15}+3060a^{14}-8568a^{13}-6189a^{12}+12376a^{11}+8019a^{10}-11441a^{9}-6481a^{8}+6443a^{7}+3094a^{6}-2024a^{5}-800a^{4}+310a^{3}+96a^{2}-21a-2$, $a^{16}-16a^{14}+104a^{12}+a^{11}-352a^{10}-11a^{9}+660a^{8}+44a^{7}-672a^{6}-77a^{5}+335a^{4}+55a^{3}-60a^{2}-12a$, $a^{23}-a^{22}-23a^{21}+22a^{20}+229a^{19}-208a^{18}-1294a^{17}+1105a^{16}+4573a^{15}-3621a^{14}-10506a^{13}+7565a^{12}+15809a^{11}-10065a^{10}-15322a^{9}+8271a^{8}+9145a^{7}-3920a^{6}-3065a^{5}+932a^{4}+473a^{3}-80a^{2}-21a+1$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}-3289a^{5}+506a^{3}-23a$, $2a^{23}-45a^{21}+a^{20}+440a^{19}-18a^{18}-2452a^{17}+136a^{16}+8585a^{15}-561a^{14}-19634a^{13}+1377a^{12}+29549a^{11}-2057a^{10}-28756a^{9}+1835a^{8}+17276a^{7}-911a^{6}-5846a^{5}+206a^{4}+933a^{3}-8a^{2}-50a-1$, $a^{23}-23a^{21}+a^{20}+231a^{19}-19a^{18}-1329a^{17}+153a^{16}+4828a^{15}-680a^{14}-11508a^{13}+1819a^{12}+18109a^{11}-2992a^{10}-18448a^{9}+2957a^{8}+11591a^{7}-1625a^{6}-4103a^{5}+410a^{4}+694a^{3}-24a^{2}-39a-1$, $a^{20}-a^{19}-19a^{18}+18a^{17}+153a^{16}-136a^{15}-680a^{14}+561a^{13}+1819a^{12}-1376a^{11}-2993a^{10}+2047a^{9}+2967a^{8}-1801a^{7}-1659a^{6}+868a^{5}+455a^{4}-196a^{3}-43a^{2}+14a$, $a^{19}-18a^{17}+135a^{15}-546a^{13}+1287a^{11}-1782a^{9}+a^{8}+1386a^{7}-8a^{6}-540a^{5}+19a^{4}+81a^{3}-13a^{2}-2a+2$, $2a^{23}+a^{22}-44a^{21}-21a^{20}+419a^{19}+190a^{18}-2262a^{17}-968a^{16}+7616a^{15}+3044a^{14}-16576a^{13}-6085a^{12}+23387a^{11}+7667a^{10}-20878a^{9}-5821a^{8}+11153a^{7}+2421a^{6}-3208a^{5}-459a^{4}+405a^{3}+27a^{2}-16a$, $2a^{23}-46a^{21}+a^{20}+461a^{19}-18a^{18}-2642a^{17}+136a^{16}+9554a^{15}-561a^{14}-22695a^{13}+1377a^{12}+35751a^{11}-2058a^{10}-36839a^{9}+1845a^{8}+23903a^{7}-946a^{6}-9086a^{5}+256a^{4}+1768a^{3}-32a^{2}-127a+1$, $2a^{23}-46a^{21}+a^{20}+461a^{19}-18a^{18}-2642a^{17}+136a^{16}+9554a^{15}-561a^{14}-22694a^{13}+1376a^{12}+35737a^{11}-2045a^{10}-36762a^{9}+1780a^{8}+23693a^{7}-791a^{6}-8794a^{5}+81a^{4}+1583a^{3}+44a^{2}-93a-3$ 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:  \( 5459008653.485176 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{24}\cdot(2\pi)^{0}\cdot 5459008653.485176 \cdot 1}{2\cdot\sqrt{161761786626698377317203521728515625}}\cr\approx \mathstrut & 0.113858567512639 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 23*x^22 + 23*x^21 + 230*x^20 - 229*x^19 - 1312*x^18 + 1294*x^17 + 4709*x^16 - 4573*x^15 - 11067*x^14 + 10506*x^13 + 17187*x^12 - 15810*x^11 - 17391*x^10 + 15333*x^9 + 11034*x^8 - 9189*x^7 - 4088*x^6 + 3142*x^5 + 784*x^4 - 528*x^3 - 64*x^2 + 32*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 - 23*x^22 + 23*x^21 + 230*x^20 - 229*x^19 - 1312*x^18 + 1294*x^17 + 4709*x^16 - 4573*x^15 - 11067*x^14 + 10506*x^13 + 17187*x^12 - 15810*x^11 - 17391*x^10 + 15333*x^9 + 11034*x^8 - 9189*x^7 - 4088*x^6 + 3142*x^5 + 784*x^4 - 528*x^3 - 64*x^2 + 32*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 - 23*x^22 + 23*x^21 + 230*x^20 - 229*x^19 - 1312*x^18 + 1294*x^17 + 4709*x^16 - 4573*x^15 - 11067*x^14 + 10506*x^13 + 17187*x^12 - 15810*x^11 - 17391*x^10 + 15333*x^9 + 11034*x^8 - 9189*x^7 - 4088*x^6 + 3142*x^5 + 784*x^4 - 528*x^3 - 64*x^2 + 32*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 - 23*x^22 + 23*x^21 + 230*x^20 - 229*x^19 - 1312*x^18 + 1294*x^17 + 4709*x^16 - 4573*x^15 - 11067*x^14 + 10506*x^13 + 17187*x^12 - 15810*x^11 - 17391*x^10 + 15333*x^9 + 11034*x^8 - 9189*x^7 - 4088*x^6 + 3142*x^5 + 784*x^4 - 528*x^3 - 64*x^2 + 32*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 $C_2\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\zeta_{15})^+\), 4.4.6125.1, 6.6.56723625.1, 6.6.300125.1, \(\Q(\zeta_{21})^+\), 8.8.3038765625.1, 12.12.3217569633140625.1, 12.12.8208085798828125.1, \(\Q(\zeta_{35})^+\)

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 ${\href{/padicField/2.12.0.1}{12} }^{2}$ R R R ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{6}$ ${\href{/padicField/17.12.0.1}{12} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{12}$ ${\href{/padicField/43.4.0.1}{4} }^{6}$ ${\href{/padicField/47.12.0.1}{12} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{8}$

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
\(3\) Copy content Toggle raw display Deg $24$$2$$12$$12$
\(5\) Copy content Toggle raw display 5.12.9.1$x^{12} - 30 x^{8} + 225 x^{4} + 1125$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.1$x^{12} - 30 x^{8} + 225 x^{4} + 1125$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(7\) Copy content Toggle raw display Deg $24$$6$$4$$20$