Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^29 - x - 3)
gp: K = bnfinit(y^29 - y - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - x - 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - x - 3)
\( x^{29} - x - 3 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $29$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(58740423215346196376309620946141855703717586307887726973\)
\(\medspace = 179\cdot 373\cdot 277813\cdot 42259019123\cdot 2243721410717\cdot 33399064272490942843393\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(83.77\) | 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: | $179^{1/2}373^{1/2}277813^{1/2}42259019123^{1/2}2243721410717^{1/2}33399064272490942843393^{1/2}\approx 7.664230112369161e+27$ | ||
| Ramified primes: |
\(179\), \(373\), \(277813\), \(42259019123\), \(2243721410717\), \(33399064272490942843393\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{58740\!\cdots\!26973}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$
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: | $14$ | 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: |
$a^{15}-a-1$, $a^{27}-a^{26}-a^{25}+2a^{24}-2a^{22}+a^{21}+a^{20}-a^{19}+a^{13}-a^{12}-2a^{11}+3a^{10}+a^{9}-4a^{8}+a^{7}+3a^{6}-2a^{5}-a^{4}+a^{3}-1$, $2a^{27}+a^{26}-2a^{25}-2a^{24}+a^{23}+2a^{22}-a^{20}-a^{17}+2a^{15}+a^{14}-2a^{13}-2a^{12}+a^{11}+2a^{10}-a^{8}-a^{5}+2a^{3}+a^{2}-3a-4$, $a^{26}+a^{25}+a^{24}-a^{19}-a^{18}+a^{16}-a^{14}-a^{13}-a^{12}-2a^{11}-3a^{10}-2a^{9}-a^{8}+a^{5}-a^{2}-1$, $4a^{28}+a^{27}-3a^{26}-5a^{25}-3a^{24}+3a^{22}+3a^{21}-a^{20}-4a^{19}-5a^{18}-2a^{17}+4a^{16}+7a^{15}+5a^{14}-2a^{13}-10a^{12}-12a^{11}-7a^{10}+4a^{9}+14a^{8}+14a^{7}+6a^{6}-8a^{5}-19a^{4}-17a^{3}-6a^{2}+9a+14$, $3a^{28}-4a^{26}+4a^{25}-5a^{24}+a^{23}+3a^{22}-6a^{21}+6a^{20}-3a^{19}-5a^{18}+6a^{17}-7a^{16}+2a^{15}+3a^{14}-6a^{13}+6a^{12}-6a^{11}-2a^{10}+5a^{9}-10a^{8}+7a^{7}-8a^{5}+9a^{4}-10a^{3}-a^{2}+6a-14$, $2a^{27}+2a^{26}+a^{25}-a^{24}-a^{23}-2a^{22}+a^{21}-a^{20}-a^{18}+2a^{16}+4a^{15}+a^{14}-5a^{12}-4a^{11}-2a^{10}+a^{9}+4a^{8}+3a^{7}+2a^{6}+2a^{5}-a^{4}-2a^{3}-4a^{2}-7a-1$, $3a^{28}-a^{27}+a^{26}-a^{24}-4a^{22}+4a^{20}-3a^{19}+a^{18}+7a^{17}-3a^{16}-3a^{15}+3a^{14}-4a^{13}-3a^{12}+a^{11}+5a^{9}+a^{8}-2a^{7}+7a^{6}-3a^{5}-11a^{4}+5a^{3}-9a+5$, $a^{28}-2a^{27}+a^{26}-a^{25}+2a^{24}+2a^{22}+a^{21}+5a^{20}-5a^{17}-4a^{16}-5a^{15}-2a^{14}-4a^{13}+2a^{12}+5a^{11}+8a^{10}+7a^{9}+3a^{8}-a^{7}-a^{6}-3a^{5}-7a^{4}-5a^{3}-5a^{2}+2a+1$, $2a^{28}+2a^{26}+a^{25}-2a^{24}+a^{23}-4a^{22}-2a^{20}+3a^{18}+4a^{16}-a^{15}-a^{14}-a^{13}-5a^{12}+a^{11}-4a^{10}+3a^{9}+3a^{8}+2a^{7}+4a^{6}-3a^{5}-6a^{3}-4a^{2}-2a-5$, $7a^{28}-8a^{27}+3a^{26}+6a^{25}-9a^{24}+6a^{23}+5a^{22}-9a^{21}+8a^{20}+4a^{19}-10a^{18}+11a^{17}+a^{16}-10a^{15}+14a^{14}-5a^{13}-8a^{12}+15a^{11}-9a^{10}-7a^{9}+14a^{8}-12a^{7}-4a^{6}+14a^{5}-18a^{4}+a^{3}+15a^{2}-21a-1$, $2a^{28}-2a^{27}+2a^{26}-a^{25}+2a^{24}-a^{23}+2a^{22}+a^{21}+3a^{20}+2a^{19}+2a^{18}+2a^{17}+a^{16}+2a^{15}-a^{14}+2a^{13}-a^{12}+2a^{11}-5a^{10}-5a^{8}+a^{7}-7a^{6}+a^{5}-6a^{4}-10a^{2}-7$, $8a^{28}+3a^{27}-2a^{26}+a^{25}+6a^{24}+2a^{23}-6a^{22}-4a^{21}+3a^{20}-11a^{18}-9a^{17}+a^{16}-2a^{15}-13a^{14}-11a^{13}+a^{12}+2a^{11}-9a^{10}-8a^{9}+6a^{8}+12a^{7}-a^{6}-4a^{5}+14a^{4}+22a^{3}+6a^{2}-a+10$, $5a^{28}-2a^{27}-6a^{26}+8a^{25}+a^{24}-12a^{23}+6a^{22}+7a^{21}-2a^{20}-9a^{19}+3a^{18}+11a^{17}-13a^{16}+12a^{14}-5a^{13}-10a^{12}+3a^{11}+16a^{10}-10a^{9}-11a^{8}+13a^{7}+a^{6}-12a^{5}+4a^{4}+19a^{3}-13a^{2}-19a+10$
| 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: | \( 46185846405204810 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 46185846405204810 \cdot 1}{2\cdot\sqrt{58740423215346196376309620946141855703717586307887726973}}\cr\approx \mathstrut & 0.900655132713095 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^29 - x - 3)
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^29 - x - 3, 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^29 - x - 3);
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^29 - x - 3);
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 8841761993739701954543616000000 |
| The 4565 conjugacy class representatives for $S_{29}$ |
| Character table for $S_{29}$ |
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 | $27{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{3}$ | $15{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | $29$ | $29$ | $22{,}\,{\href{/padicField/13.7.0.1}{7} }$ | $15{,}\,{\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $29$ | $18{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $29$ | $29$ | $23{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | $26{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | $24{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(179\)
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 179.2.1.1 | $x^{2} + 358$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 179.4.0.1 | $x^{4} + x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 179.5.0.1 | $x^{5} + 2 x + 177$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 179.17.0.1 | $x^{17} + 4 x + 177$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
|
\(373\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(277813\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
|
\(42259019123\)
| $\Q_{42259019123}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
|
\(2243721410717\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
|
\(333\!\cdots\!393\)
| $\Q_{33\!\cdots\!93}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ |