Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 139*x^6 + 23*x^5 + 6019*x^4 + 7188*x^3 - 83676*x^2 - 171672*x + 62576)
gp: K = bnfinit(y^8 - 2*y^7 - 139*y^6 + 23*y^5 + 6019*y^4 + 7188*y^3 - 83676*y^2 - 171672*y + 62576, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^7 - 139*x^6 + 23*x^5 + 6019*x^4 + 7188*x^3 - 83676*x^2 - 171672*x + 62576);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 2*x^7 - 139*x^6 + 23*x^5 + 6019*x^4 + 7188*x^3 - 83676*x^2 - 171672*x + 62576)
\( x^{8} - 2x^{7} - 139x^{6} + 23x^{5} + 6019x^{4} + 7188x^{3} - 83676x^{2} - 171672x + 62576 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(3226963558964161\)
\(\medspace = 7537^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(86.82\) | 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: | $7537^{1/2}\approx 86.81589716175259$ | ||
| Ramified primes: |
\(7537\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{148}a^{6}-\frac{5}{37}a^{5}-\frac{63}{148}a^{4}+\frac{29}{148}a^{3}+\frac{5}{148}a^{2}+\frac{23}{74}a+\frac{16}{37}$, $\frac{1}{25219700536}a^{7}+\frac{23202527}{12609850268}a^{6}+\frac{4102474845}{25219700536}a^{5}+\frac{10686037631}{25219700536}a^{4}+\frac{8810468587}{25219700536}a^{3}+\frac{644545487}{6304925134}a^{2}+\frac{1111735267}{6304925134}a+\frac{850205444}{3152462567}$
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
$C_{3}$, which has order $3$ (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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{21381919}{25219700536}a^{7}-\frac{68689453}{12609850268}a^{6}-\frac{2340740481}{25219700536}a^{5}+\frac{10834583109}{25219700536}a^{4}+\frac{78219072605}{25219700536}a^{3}-\frac{49092654689}{6304925134}a^{2}-\frac{108146662329}{3152462567}a+\frac{37046979240}{3152462567}$, $\frac{495918}{3152462567}a^{7}-\frac{5944129}{6304925134}a^{6}-\frac{114024731}{6304925134}a^{5}+\frac{490416157}{6304925134}a^{4}+\frac{1999762096}{3152462567}a^{3}-\frac{4842954428}{3152462567}a^{2}-\frac{45790990983}{6304925134}a+\frac{15219770544}{3152462567}$, $\frac{6447101}{6304925134}a^{7}-\frac{84693403}{12609850268}a^{6}-\frac{346029061}{3152462567}a^{5}+\frac{6583546187}{12609850268}a^{4}+\frac{45092981825}{12609850268}a^{3}-\frac{114690912221}{12609850268}a^{2}-\frac{122120681189}{3152462567}a+\frac{40462912896}{3152462567}$, $\frac{29069733}{25219700536}a^{7}-\frac{83430637}{12609850268}a^{6}-\frac{3082065275}{25219700536}a^{5}+\frac{12247877219}{25219700536}a^{4}+\frac{93692969099}{25219700536}a^{3}-\frac{25943706184}{3152462567}a^{2}-\frac{116122662947}{3152462567}a+\frac{40968975652}{3152462567}$, $\frac{78607813}{12609850268}a^{7}-\frac{256246663}{6304925134}a^{6}-\frac{8652387277}{12609850268}a^{5}+\frac{40978093607}{12609850268}a^{4}+\frac{293246240789}{12609850268}a^{3}-\frac{377609892561}{6304925134}a^{2}-\frac{1659609687487}{6304925134}a+\frac{279612630814}{3152462567}$, $\frac{17338331}{25219700536}a^{7}+\frac{17642141}{12609850268}a^{6}-\frac{2277598941}{25219700536}a^{5}-\frac{8138268723}{25219700536}a^{4}+\frac{71864766381}{25219700536}a^{3}+\frac{42233469362}{3152462567}a^{2}-\frac{24665664899}{3152462567}a-\frac{210345926686}{3152462567}$, $\frac{15385423388615}{6304925134}a^{7}-\frac{103688607807583}{6304925134}a^{6}-\frac{821904870754252}{3152462567}a^{5}+\frac{40\!\cdots\!13}{3152462567}a^{4}+\frac{53\!\cdots\!03}{6304925134}a^{3}-\frac{71\!\cdots\!63}{3152462567}a^{2}-\frac{59\!\cdots\!63}{6304925134}a+\frac{93\!\cdots\!54}{3152462567}$
| 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: | \( 338576.421925 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 338576.421925 \cdot 3}{2\cdot\sqrt{3226963558964161}}\cr\approx \mathstrut & 2.28871072571 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 139*x^6 + 23*x^5 + 6019*x^4 + 7188*x^3 - 83676*x^2 - 171672*x + 62576)
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^8 - 2*x^7 - 139*x^6 + 23*x^5 + 6019*x^4 + 7188*x^3 - 83676*x^2 - 171672*x + 62576, 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^8 - 2*x^7 - 139*x^6 + 23*x^5 + 6019*x^4 + 7188*x^3 - 83676*x^2 - 171672*x + 62576);
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^8 - 2*x^7 - 139*x^6 + 23*x^5 + 6019*x^4 + 7188*x^3 - 83676*x^2 - 171672*x + 62576);
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
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)
| A solvable group of order 24 |
| The 5 conjugacy class representatives for $S_4$ |
| Character table for $S_4$ |
Intermediate fields
| \(\Q(\sqrt{7537}) \), 4.4.7537.1 |
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]
Sibling fields
| Galois closure: | deg 24 |
| Degree 4 sibling: | 4.4.7537.1 |
| Degree 6 siblings: | 6.6.56806369.1, 6.6.428149603153.2 |
| Degree 12 siblings: | deg 12, deg 12 |
| Minimal sibling: | 4.4.7537.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7537\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |