Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^37 - x - 4)
gp: K = bnfinit(y^37 - y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^37 - x - 4);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^37 - x - 4)
\( x^{37} - x - 4 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $37$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(725343351038911776109400911086491093773761406947891105827382662004736\)
\(\medspace = 2^{38}\cdot 40158589\cdot 859997730921167\cdot 76\!\cdots\!63\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(72.63\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(40158589\), \(859997730921167\), \(76406\!\cdots\!91263\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{26387\!\cdots\!67469}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2}a^{19}-\frac{1}{2}a$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{11}$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{12}$, $\frac{1}{2}a^{31}-\frac{1}{2}a^{13}$, $\frac{1}{2}a^{32}-\frac{1}{2}a^{14}$, $\frac{1}{2}a^{33}-\frac{1}{2}a^{15}$, $\frac{1}{2}a^{34}-\frac{1}{2}a^{16}$, $\frac{1}{2}a^{35}-\frac{1}{2}a^{17}$, $\frac{1}{2}a^{36}-\frac{1}{2}a^{18}$
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
not computed
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: | $18$ | 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: | not computed | 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: | not computed | 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 = \mathstrut &\frac{2^{1}\cdot(2\pi)^{18}\cdot R \cdot h}{2\cdot\sqrt{725343351038911776109400911086491093773761406947891105827382662004736}}\cr\mathstrut & \text{
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^37 - x - 4)
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^37 - x - 4, 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^37 - x - 4);
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^37 - x - 4);
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 non-solvable group of order 13763753091226345046315979581580902400000000 |
| The 21637 conjugacy class representatives for $S_{37}$ are not computed |
| Character table for $S_{37}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 | $29{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | $28{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | $37$ | $33{,}\,{\href{/padicField/11.4.0.1}{4} }$ | $37$ | $28{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.6.0.1}{6} }$ | $25{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.5.0.1}{5} }$ | $21{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $29{,}\,{\href{/padicField/31.8.0.1}{8} }$ | $37$ | $32{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $17{,}\,15{,}\,{\href{/padicField/47.5.0.1}{5} }$ | $31{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | $21{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.4.6.3 | $x^{4} + 8 x^{3} + 28 x^{2} + 48 x + 84$ | $2$ | $2$ | $6$ | $C_4$ | $[3]^{2}$ | |
| 2.8.8.7 | $x^{8} + 8 x^{7} + 40 x^{6} + 120 x^{5} + 232 x^{4} + 240 x^{3} + 160 x^{2} + 64 x + 16$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ | |
| 2.12.12.19 | $x^{12} + 6 x^{11} + 40 x^{10} + 188 x^{9} + 732 x^{8} + 2896 x^{7} + 8224 x^{6} + 22240 x^{5} + 43760 x^{4} + 56672 x^{3} + 77824 x^{2} - 19776 x + 66112$ | $2$ | $6$ | $12$ | 12T105 | $[2, 2, 2, 2]^{12}$ | |
| 2.12.12.16 | $x^{12} - 6 x^{11} + 16 x^{10} - 12 x^{9} - 68 x^{8} + 16 x^{7} + 832 x^{6} - 4320 x^{5} + 11632 x^{4} - 6240 x^{3} + 13568 x^{2} + 9536 x + 1600$ | $2$ | $6$ | $12$ | 12T134 | $[2, 2, 2, 2, 2, 2]^{6}$ | |
|
\(40158589\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $30$ | $1$ | $30$ | $0$ | $C_{30}$ | $[\ ]^{30}$ | ||
|
\(859997730921167\)
| $\Q_{859997730921167}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(764\!\cdots\!263\)
| $\Q_{76\!\cdots\!63}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $29$ | $1$ | $29$ | $0$ | $C_{29}$ | $[\ ]^{29}$ |