Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 7*x^22 + 35*x^20 + 154*x^18 + 637*x^16 + 1666*x^14 + 3822*x^12 + 7889*x^10 + 13377*x^8 + 9947*x^6 + 7203*x^4 + 4802*x^2 + 2401)
gp: K = bnfinit(y^24 + 7*y^22 + 35*y^20 + 154*y^18 + 637*y^16 + 1666*y^14 + 3822*y^12 + 7889*y^10 + 13377*y^8 + 9947*y^6 + 7203*y^4 + 4802*y^2 + 2401, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 7*x^22 + 35*x^20 + 154*x^18 + 637*x^16 + 1666*x^14 + 3822*x^12 + 7889*x^10 + 13377*x^8 + 9947*x^6 + 7203*x^4 + 4802*x^2 + 2401);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 + 7*x^22 + 35*x^20 + 154*x^18 + 637*x^16 + 1666*x^14 + 3822*x^12 + 7889*x^10 + 13377*x^8 + 9947*x^6 + 7203*x^4 + 4802*x^2 + 2401)
\( x^{24} + 7 x^{22} + 35 x^{20} + 154 x^{18} + 637 x^{16} + 1666 x^{14} + 3822 x^{12} + 7889 x^{10} + \cdots + 2401 \)
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: |
\(5106705043047168064000000000000000000\)
\(\medspace = 2^{24}\cdot 5^{18}\cdot 7^{20}\)
| 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: |
\(2\), \(5\), \(7\)
| 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$$\chi_{140}(19,·)$, $\chi_{140}(1,·)$, $\chi_{140}(3,·)$, $\chi_{140}(9,·)$, $\chi_{140}(139,·)$, $\chi_{140}(81,·)$, $\chi_{140}(131,·)$, $\chi_{140}(121,·)$, $\chi_{140}(87,·)$, $\chi_{140}(27,·)$, $\chi_{140}(29,·)$, $\chi_{140}(31,·)$, $\chi_{140}(37,·)$, $\chi_{140}(103,·)$, $\chi_{140}(109,·)$, $\chi_{140}(93,·)$, $\chi_{140}(47,·)$, $\chi_{140}(113,·)$, $\chi_{140}(83,·)$, $\chi_{140}(53,·)$, $\chi_{140}(137,·)$, $\chi_{140}(111,·)$, $\chi_{140}(57,·)$, $\chi_{140}(59,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}$, $\frac{1}{7}a^{7}$, $\frac{1}{7}a^{8}$, $\frac{1}{7}a^{9}$, $\frac{1}{7}a^{10}$, $\frac{1}{7}a^{11}$, $\frac{1}{49}a^{12}$, $\frac{1}{49}a^{13}$, $\frac{1}{49}a^{14}$, $\frac{1}{49}a^{15}$, $\frac{1}{49}a^{16}$, $\frac{1}{49}a^{17}$, $\frac{1}{998473}a^{18}-\frac{661}{142639}a^{16}-\frac{1014}{142639}a^{14}-\frac{754}{142639}a^{12}+\frac{613}{20377}a^{10}+\frac{928}{20377}a^{8}-\frac{131}{20377}a^{6}-\frac{739}{2911}a^{4}-\frac{1072}{2911}a^{2}-\frac{200}{2911}$, $\frac{1}{998473}a^{19}-\frac{661}{142639}a^{17}-\frac{1014}{142639}a^{15}-\frac{754}{142639}a^{13}+\frac{613}{20377}a^{11}+\frac{928}{20377}a^{9}-\frac{131}{20377}a^{7}-\frac{739}{2911}a^{5}-\frac{1072}{2911}a^{3}-\frac{200}{2911}a$, $\frac{1}{998473}a^{20}-\frac{946}{20377}a^{10}+\frac{298}{2911}$, $\frac{1}{998473}a^{21}-\frac{946}{20377}a^{11}+\frac{298}{2911}a$, $\frac{1}{998473}a^{22}-\frac{800}{142639}a^{12}+\frac{298}{2911}a^{2}$, $\frac{1}{998473}a^{23}-\frac{800}{142639}a^{13}+\frac{298}{2911}a^{3}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{26}$, which has order $26$ (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: |
\( \frac{311}{998473} a^{22} + \frac{311}{142639} a^{20} + \frac{10907}{998473} a^{18} + \frac{6842}{142639} a^{16} + \frac{4043}{20377} a^{14} + \frac{10574}{20377} a^{12} + \frac{24258}{20377} a^{10} + \frac{49636}{20377} a^{8} + \frac{12129}{2911} a^{6} + \frac{9019}{2911} a^{4} + \frac{6531}{2911} a^{2} + \frac{4354}{2911} \)
(order $10$)
| 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: |
$\frac{13}{998473}a^{22}-\frac{1667}{142639}a^{12}+\frac{3874}{2911}a^{2}+1$, $\frac{26}{998473}a^{22}-\frac{3334}{142639}a^{12}+\frac{4837}{2911}a^{2}$, $\frac{116}{998473}a^{22}+\frac{580}{998473}a^{20}+\frac{2552}{998473}a^{18}+\frac{1508}{142639}a^{16}+\frac{5973}{142639}a^{14}+\frac{9048}{142639}a^{12}+\frac{2668}{20377}a^{10}+\frac{4524}{20377}a^{8}+\frac{3364}{20377}a^{6}-\frac{2511}{2911}a^{4}+\frac{232}{2911}a^{2}+\frac{116}{2911}$, $\frac{298}{998473}a^{22}+\frac{30}{14063}a^{20}+\frac{10907}{998473}a^{18}+\frac{6842}{142639}a^{16}+\frac{4043}{20377}a^{14}+\frac{75685}{142639}a^{12}+\frac{25055}{20377}a^{10}+\frac{49636}{20377}a^{8}+\frac{12129}{2911}a^{6}+\frac{9019}{2911}a^{4}+\frac{2657}{2911}a^{2}-\frac{919}{2911}$, $\frac{583}{998473}a^{22}+\frac{3874}{998473}a^{20}+\frac{2682}{142639}a^{18}+\frac{11623}{142639}a^{16}+\frac{47680}{142639}a^{14}+\frac{116397}{142639}a^{12}+\frac{36356}{20377}a^{10}+\frac{72712}{20377}a^{8}+\frac{113930}{20377}a^{6}+\frac{5662}{2911}a^{4}-\frac{825}{2911}$, $\frac{458}{998473}a^{23}+\frac{4}{998473}a^{22}+\frac{2977}{998473}a^{21}+\frac{2061}{142639}a^{19}+\frac{8922}{142639}a^{17}+\frac{36640}{142639}a^{15}+\frac{12595}{20377}a^{13}-\frac{289}{142639}a^{12}+\frac{27938}{20377}a^{11}+\frac{55876}{20377}a^{9}+\frac{89543}{20377}a^{7}+\frac{4351}{2911}a^{5}+\frac{2977}{2911}a^{3}-\frac{1719}{2911}a^{2}+\frac{1603}{2911}a$, $\frac{39}{998473}a^{23}-\frac{704}{998473}a^{22}-\frac{704}{142639}a^{20}-\frac{24643}{998473}a^{18}-\frac{15488}{142639}a^{16}-\frac{9152}{20377}a^{14}-\frac{5001}{142639}a^{13}-\frac{23936}{20377}a^{12}-\frac{54912}{20377}a^{10}-\frac{113417}{20377}a^{8}-\frac{27456}{2911}a^{6}-\frac{20416}{2911}a^{4}+\frac{8711}{2911}a^{3}-\frac{14784}{2911}a^{2}-\frac{9856}{2911}$, $\frac{345}{998473}a^{23}+\frac{1725}{998473}a^{21}+\frac{25}{998473}a^{20}+\frac{7590}{998473}a^{19}+\frac{4485}{142639}a^{17}+\frac{17890}{142639}a^{15}+\frac{26910}{142639}a^{13}+\frac{7935}{20377}a^{11}-\frac{362}{20377}a^{10}+\frac{13455}{20377}a^{9}+\frac{10005}{20377}a^{7}-\frac{8246}{2911}a^{5}+\frac{690}{2911}a^{3}+\frac{345}{2911}a-\frac{1283}{2911}$, $\frac{872}{998473}a^{23}+\frac{3}{20377}a^{22}+\frac{5668}{998473}a^{21}+\frac{800}{998473}a^{20}+\frac{3924}{142639}a^{19}+\frac{3520}{998473}a^{18}+\frac{17025}{142639}a^{17}+\frac{2080}{142639}a^{16}+\frac{69760}{142639}a^{15}+\frac{8339}{142639}a^{14}+\frac{23980}{20377}a^{13}+\frac{2021}{20377}a^{12}+\frac{53192}{20377}a^{11}+\frac{3680}{20377}a^{10}+\frac{106384}{20377}a^{9}+\frac{6240}{20377}a^{8}+\frac{165615}{20377}a^{7}+\frac{4640}{20377}a^{6}+\frac{8284}{2911}a^{5}-\frac{4668}{2911}a^{4}+\frac{5668}{2911}a^{3}-\frac{3554}{2911}a^{2}+\frac{3052}{2911}a-\frac{2751}{2911}$, $\frac{39}{998473}a^{23}+\frac{3}{20377}a^{22}+\frac{800}{998473}a^{20}+\frac{3520}{998473}a^{18}+\frac{2080}{142639}a^{16}+\frac{8339}{142639}a^{14}-\frac{5001}{142639}a^{13}+\frac{2021}{20377}a^{12}+\frac{3680}{20377}a^{10}+\frac{6240}{20377}a^{8}+\frac{4640}{20377}a^{6}-\frac{4668}{2911}a^{4}+\frac{8711}{2911}a^{3}-\frac{3554}{2911}a^{2}+\frac{160}{2911}$, $\frac{458}{998473}a^{23}-\frac{3}{20377}a^{22}+\frac{2977}{998473}a^{21}-\frac{800}{998473}a^{20}+\frac{2061}{142639}a^{19}-\frac{3520}{998473}a^{18}+\frac{8922}{142639}a^{17}-\frac{2080}{142639}a^{16}+\frac{36640}{142639}a^{15}-\frac{8339}{142639}a^{14}+\frac{12595}{20377}a^{13}-\frac{2021}{20377}a^{12}+\frac{27938}{20377}a^{11}-\frac{3680}{20377}a^{10}+\frac{55876}{20377}a^{9}-\frac{6240}{20377}a^{8}+\frac{89543}{20377}a^{7}-\frac{4640}{20377}a^{6}+\frac{4351}{2911}a^{5}+\frac{4668}{2911}a^{4}+\frac{2977}{2911}a^{3}+\frac{3554}{2911}a^{2}+\frac{1603}{2911}a+\frac{2751}{2911}$
| 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: | \( 24838443.50460296 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 24838443.50460296 \cdot 26}{10\cdot\sqrt{5106705043047168064000000000000000000}}\cr\approx \mathstrut & 0.108189660637048 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 + 7*x^22 + 35*x^20 + 154*x^18 + 637*x^16 + 1666*x^14 + 3822*x^12 + 7889*x^10 + 13377*x^8 + 9947*x^6 + 7203*x^4 + 4802*x^2 + 2401)
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 + 7*x^22 + 35*x^20 + 154*x^18 + 637*x^16 + 1666*x^14 + 3822*x^12 + 7889*x^10 + 13377*x^8 + 9947*x^6 + 7203*x^4 + 4802*x^2 + 2401, 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 + 7*x^22 + 35*x^20 + 154*x^18 + 637*x^16 + 1666*x^14 + 3822*x^12 + 7889*x^10 + 13377*x^8 + 9947*x^6 + 7203*x^4 + 4802*x^2 + 2401);
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 + 7*x^22 + 35*x^20 + 154*x^18 + 637*x^16 + 1666*x^14 + 3822*x^12 + 7889*x^10 + 13377*x^8 + 9947*x^6 + 7203*x^4 + 4802*x^2 + 2401);
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
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 | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{6}$ | ${\href{/padicField/17.12.0.1}{12} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.3.0.1}{3} }^{8}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 $24$ | $2$ | $12$ | $24$ | |||
|
\(5\)
| Deg $24$ | $4$ | $6$ | $18$ | |||
|
\(7\)
| Deg $24$ | $6$ | $4$ | $20$ |