Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 31*x^20 + 317*x^16 + 1216*x^12 + 1462*x^8 + 301*x^4 + 1)
gp: K = bnfinit(y^24 + 31*y^20 + 317*y^16 + 1216*y^12 + 1462*y^8 + 301*y^4 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 31*x^20 + 317*x^16 + 1216*x^12 + 1462*x^8 + 301*x^4 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 + 31*x^20 + 317*x^16 + 1216*x^12 + 1462*x^8 + 301*x^4 + 1)
\( x^{24} + 31x^{20} + 317x^{16} + 1216x^{12} + 1462x^{8} + 301x^{4} + 1 \)
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: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(5349421735921433961299014349756563456\)
\(\medspace = 2^{48}\cdot 13^{20}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.91\) | 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: | $2^{2}13^{5/6}\approx 33.911431129417196$ | ||
| Ramified primes: |
\(2\), \(13\)
| 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: | \(104=2^{3}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{104}(1,·)$, $\chi_{104}(3,·)$, $\chi_{104}(69,·)$, $\chi_{104}(103,·)$, $\chi_{104}(9,·)$, $\chi_{104}(75,·)$, $\chi_{104}(77,·)$, $\chi_{104}(79,·)$, $\chi_{104}(17,·)$, $\chi_{104}(23,·)$, $\chi_{104}(25,·)$, $\chi_{104}(27,·)$, $\chi_{104}(29,·)$, $\chi_{104}(95,·)$, $\chi_{104}(35,·)$, $\chi_{104}(101,·)$, $\chi_{104}(81,·)$, $\chi_{104}(43,·)$, $\chi_{104}(49,·)$, $\chi_{104}(51,·)$, $\chi_{104}(53,·)$, $\chi_{104}(55,·)$, $\chi_{104}(87,·)$, $\chi_{104}(61,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5}a^{16}+\frac{1}{5}a^{8}+\frac{1}{5}$, $\frac{1}{5}a^{17}+\frac{1}{5}a^{9}+\frac{1}{5}a$, $\frac{1}{5}a^{18}+\frac{1}{5}a^{10}+\frac{1}{5}a^{2}$, $\frac{1}{5}a^{19}+\frac{1}{5}a^{11}+\frac{1}{5}a^{3}$, $\frac{1}{5758715}a^{20}+\frac{22879}{1151743}a^{16}+\frac{2324846}{5758715}a^{12}-\frac{236478}{1151743}a^{8}+\frac{993646}{5758715}a^{4}+\frac{121670}{1151743}$, $\frac{1}{5758715}a^{21}+\frac{22879}{1151743}a^{17}+\frac{2324846}{5758715}a^{13}-\frac{236478}{1151743}a^{9}+\frac{993646}{5758715}a^{5}+\frac{121670}{1151743}a$, $\frac{1}{5758715}a^{22}+\frac{22879}{1151743}a^{18}+\frac{2324846}{5758715}a^{14}-\frac{236478}{1151743}a^{10}+\frac{993646}{5758715}a^{6}+\frac{121670}{1151743}a^{2}$, $\frac{1}{5758715}a^{23}+\frac{22879}{1151743}a^{19}+\frac{2324846}{5758715}a^{15}-\frac{236478}{1151743}a^{11}+\frac{993646}{5758715}a^{7}+\frac{121670}{1151743}a^{3}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
Class group and class number
$C_{3}\times C_{9}$, which has order $27$ (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: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{125684}{5758715} a^{21} + \frac{773708}{1151743} a^{17} + \frac{39056569}{5758715} a^{13} + \frac{29172481}{1151743} a^{9} + \frac{163154394}{5758715} a^{5} + \frac{4887441}{1151743} a \)
(order $8$)
| 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: |
$\frac{74816}{5758715}a^{20}+\frac{2277573}{5758715}a^{16}+\frac{22485336}{5758715}a^{12}+\frac{79704628}{5758715}a^{8}+\frac{76230496}{5758715}a^{4}+\frac{4340698}{5758715}$, $\frac{5257}{1151743}a^{21}+\frac{164669}{1151743}a^{17}+\frac{1722192}{1151743}a^{13}+\frac{7043199}{1151743}a^{9}+\frac{10808204}{1151743}a^{5}+\frac{5464354}{1151743}a$, $\frac{71896}{5758715}a^{21}+\frac{2249643}{5758715}a^{17}+\frac{23460001}{5758715}a^{13}+\frac{94330573}{5758715}a^{9}+\frac{129004971}{5758715}a^{5}+\frac{36195208}{5758715}a$, $\frac{262819}{5758715}a^{22}+\frac{8142434}{5758715}a^{18}+\frac{83143954}{5758715}a^{14}+\frac{317405974}{5758715}a^{10}+\frac{371398014}{5758715}a^{6}+\frac{50500339}{5758715}a^{2}$, $\frac{71896}{5758715}a^{22}+\frac{2249643}{5758715}a^{18}+\frac{23460001}{5758715}a^{14}+\frac{94330573}{5758715}a^{10}+\frac{129004971}{5758715}a^{6}+\frac{36195208}{5758715}a^{2}$, $\frac{1488794}{5758715}a^{23}-\frac{294361}{5758715}a^{22}+\frac{125684}{5758715}a^{21}+\frac{46118454}{5758715}a^{19}-\frac{9130448}{5758715}a^{18}+\frac{773708}{1151743}a^{17}+\frac{470926754}{5758715}a^{15}-\frac{93477106}{5758715}a^{14}+\frac{39056569}{5758715}a^{13}+\frac{1800555279}{5758715}a^{11}-\frac{359665168}{5758715}a^{10}+\frac{29172481}{1151743}a^{9}+\frac{2143183414}{5758715}a^{7}-\frac{437398981}{5758715}a^{6}+\frac{163154394}{5758715}a^{5}+\frac{418252269}{5758715}a^{3}-\frac{100562608}{5758715}a^{2}+\frac{4887441}{1151743}a$, $\frac{1488794}{5758715}a^{23}-\frac{125684}{5758715}a^{21}-\frac{70496}{5758715}a^{20}+\frac{46118454}{5758715}a^{19}-\frac{773708}{1151743}a^{17}-\frac{437784}{1151743}a^{16}+\frac{470926754}{5758715}a^{15}-\frac{39056569}{5758715}a^{13}-\frac{22349576}{5758715}a^{12}+\frac{1800555279}{5758715}a^{11}-\frac{29172481}{1151743}a^{9}-\frac{16851239}{1151743}a^{8}+\frac{2143183414}{5758715}a^{7}-\frac{163154394}{5758715}a^{5}-\frac{91198596}{5758715}a^{4}+\frac{418252269}{5758715}a^{3}-\frac{4887441}{1151743}a-\frac{1369942}{1151743}$, $\frac{294361}{5758715}a^{22}+\frac{9130448}{5758715}a^{18}+\frac{93477106}{5758715}a^{14}+\frac{359665168}{5758715}a^{10}+\frac{437398981}{5758715}a^{6}+\frac{100562608}{5758715}a^{2}-a+1$, $\frac{1488794}{5758715}a^{23}+\frac{111436}{1151743}a^{22}+\frac{125684}{5758715}a^{21}+\frac{46118454}{5758715}a^{19}+\frac{17272882}{5758715}a^{18}+\frac{773708}{1151743}a^{17}+\frac{470926754}{5758715}a^{15}+\frac{35324212}{1151743}a^{14}+\frac{39056569}{5758715}a^{13}+\frac{1800555279}{5758715}a^{11}+\frac{677071142}{5758715}a^{10}+\frac{29172481}{1151743}a^{9}+\frac{2143183414}{5758715}a^{7}+\frac{161759399}{1151743}a^{6}+\frac{163154394}{5758715}a^{5}+\frac{418252269}{5758715}a^{3}+\frac{151062947}{5758715}a^{2}+\frac{4887441}{1151743}a$, $\frac{1488794}{5758715}a^{23}-\frac{125684}{5758715}a^{21}+\frac{21028}{5758715}a^{20}+\frac{46118454}{5758715}a^{19}-\frac{773708}{1151743}a^{17}+\frac{658676}{5758715}a^{16}+\frac{470926754}{5758715}a^{15}-\frac{39056569}{5758715}a^{13}+\frac{6888768}{5758715}a^{12}+\frac{1800555279}{5758715}a^{11}-\frac{29172481}{1151743}a^{9}+\frac{28172796}{5758715}a^{8}+\frac{2143183414}{5758715}a^{7}-\frac{163154394}{5758715}a^{5}+\frac{42081073}{5758715}a^{4}+\frac{418252269}{5758715}a^{3}-\frac{4887441}{1151743}a+\frac{10339986}{5758715}$, $\frac{1488794}{5758715}a^{23}+\frac{588722}{5758715}a^{22}+\frac{125684}{5758715}a^{21}+\frac{46118454}{5758715}a^{19}+\frac{18260896}{5758715}a^{18}+\frac{773708}{1151743}a^{17}+\frac{470926754}{5758715}a^{15}+\frac{186954212}{5758715}a^{14}+\frac{39056569}{5758715}a^{13}+\frac{1800555279}{5758715}a^{11}+\frac{719330336}{5758715}a^{10}+\frac{29172481}{1151743}a^{9}+\frac{2143183414}{5758715}a^{7}+\frac{874797962}{5758715}a^{6}+\frac{163154394}{5758715}a^{5}+\frac{418252269}{5758715}a^{3}+\frac{195366501}{5758715}a^{2}+\frac{4887441}{1151743}a$
| 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: | \( 34260599.803001165 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 34260599.803001165 \cdot 27}{8\cdot\sqrt{5349421735921433961299014349756563456}}\cr\approx \mathstrut & 0.189266495358777 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 + 31*x^20 + 317*x^16 + 1216*x^12 + 1462*x^8 + 301*x^4 + 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 + 31*x^20 + 317*x^16 + 1216*x^12 + 1462*x^8 + 301*x^4 + 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 + 31*x^20 + 317*x^16 + 1216*x^12 + 1462*x^8 + 301*x^4 + 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 + 31*x^20 + 317*x^16 + 1216*x^12 + 1462*x^8 + 301*x^4 + 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
$C_2^2\times C_6$ (as 24T3):
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 $C_2^2\times C_6$ |
| Character table for $C_2^2\times C_6$ |
Intermediate fields
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 | ${\href{/padicField/3.6.0.1}{6} }^{4}$ | ${\href{/padicField/5.2.0.1}{2} }^{12}$ | ${\href{/padicField/7.6.0.1}{6} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{8}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{12}$ | ${\href{/padicField/37.6.0.1}{6} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{12}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $24$ | $4$ | $6$ | $48$ | |||
|
\(13\)
| 13.12.10.1 | $x^{12} + 72 x^{11} + 2172 x^{10} + 35280 x^{9} + 328380 x^{8} + 1703232 x^{7} + 4282170 x^{6} + 3407400 x^{5} + 1340820 x^{4} + 712800 x^{3} + 3855192 x^{2} + 18082080 x + 35650393$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
| 13.12.10.1 | $x^{12} + 72 x^{11} + 2172 x^{10} + 35280 x^{9} + 328380 x^{8} + 1703232 x^{7} + 4282170 x^{6} + 3407400 x^{5} + 1340820 x^{4} + 712800 x^{3} + 3855192 x^{2} + 18082080 x + 35650393$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |