Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^29 + 3*x - 2)
gp: K = bnfinit(y^29 + 3*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 + 3*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 + 3*x - 2)
\( x^{29} + 3x - 2 \)
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: |
\(2275479017222419439026339091644104204888156487918026752\)
\(\medspace = 2^{28}\cdot 8447\cdot 10\!\cdots\!11\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(74.88\) | 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: |
\(2\), \(8447\), \(10035\!\cdots\!53811\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{84768\!\cdots\!41517}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{28}+a^{25}+2a^{24}+2a^{23}+a^{22}+a^{19}+2a^{18}+2a^{17}+a^{16}+a^{13}+2a^{12}+2a^{11}+a^{10}+a^{7}+2a^{6}+2a^{5}+a^{4}+a+5$, $3a^{28}+2a^{27}+2a^{26}+2a^{25}+a^{24}+2a^{23}+2a^{21}-a^{20}+2a^{19}-2a^{18}+2a^{17}-3a^{16}+3a^{15}-4a^{14}+4a^{13}-5a^{12}+5a^{11}-7a^{10}+6a^{9}-8a^{8}+6a^{7}-9a^{6}+6a^{5}-9a^{4}+4a^{3}-8a^{2}+2a+3$, $a^{27}-a^{26}-2a^{24}+2a^{21}+a^{20}+a^{19}-2a^{18}-a^{17}-2a^{16}+a^{15}+4a^{13}-a^{12}+a^{11}-3a^{10}-3a^{8}+2a^{7}+2a^{6}+a^{5}+a^{4}-2a^{3}+a^{2}-5a+3$, $2a^{28}+a^{27}+a^{26}-a^{21}+a^{17}+a^{15}+a^{14}+a^{13}+2a^{11}+a^{10}+a^{9}+2a^{8}+2a^{6}+a^{5}+a^{4}+3a^{2}+7$, $2a^{27}+a^{26}-3a^{24}-2a^{23}+3a^{22}+a^{21}-2a^{20}-3a^{19}-3a^{18}+2a^{17}+2a^{16}-2a^{15}-4a^{14}-4a^{13}+3a^{12}+3a^{11}-4a^{10}-3a^{9}-3a^{8}+a^{7}+6a^{6}-3a^{5}-6a^{4}-a^{3}+3a^{2}+6a-3$, $a^{27}-a^{26}-a^{23}+a^{20}-2a^{19}+a^{17}+a^{16}-a^{15}+a^{13}+2a^{9}-3a^{8}+a^{7}+a^{5}-3a^{4}-a^{3}+4a^{2}-3a+1$, $2a^{28}-a^{27}-7a^{26}-3a^{25}+4a^{24}+2a^{23}+4a^{22}+7a^{21}-2a^{20}-9a^{19}-3a^{18}-2a^{17}-3a^{16}+8a^{15}+11a^{14}-2a^{13}-3a^{12}-a^{11}-11a^{10}-7a^{9}+10a^{8}+8a^{7}+9a^{5}-a^{4}-18a^{3}-5a^{2}+7a+3$, $14a^{28}+8a^{27}+6a^{26}+5a^{25}+2a^{24}+3a^{22}+4a^{21}-3a^{20}-3a^{19}+6a^{18}+3a^{17}-6a^{16}-3a^{15}+7a^{14}+4a^{13}-9a^{12}-3a^{11}+10a^{10}+a^{9}-9a^{8}-a^{7}+8a^{6}+a^{5}-9a^{4}+a^{3}+7a^{2}-4a+39$, $12a^{28}-9a^{27}+3a^{26}+3a^{25}-9a^{24}+15a^{23}-19a^{22}+23a^{21}-24a^{20}+23a^{19}-18a^{18}+10a^{17}-11a^{15}+21a^{14}-29a^{13}+37a^{12}-40a^{11}+41a^{10}-35a^{9}+24a^{8}-8a^{7}-10a^{6}+28a^{5}-44a^{4}+57a^{3}-66a^{2}+71a-29$, $2a^{28}+a^{27}+3a^{26}+4a^{25}-a^{23}+5a^{22}+3a^{21}-7a^{20}-a^{19}+3a^{18}-3a^{17}-7a^{16}-a^{14}-2a^{13}+3a^{12}-3a^{11}-a^{10}+8a^{9}+7a^{8}-9a^{7}+3a^{6}+13a^{5}-3a^{4}-8a^{3}+3a^{2}+4a-3$, $13a^{28}+9a^{27}+4a^{26}-2a^{25}-8a^{24}-12a^{23}-15a^{22}-16a^{21}-15a^{20}-12a^{19}-7a^{18}-a^{17}+5a^{16}+11a^{15}+15a^{14}+18a^{13}+19a^{12}+17a^{11}+13a^{10}+7a^{9}-6a^{7}-13a^{6}-18a^{5}-21a^{4}-22a^{3}-19a^{2}-14a+31$, $a^{28}-3a^{27}-4a^{26}-a^{25}-2a^{24}-a^{23}+3a^{22}+2a^{21}-a^{20}-2a^{19}-2a^{18}-2a^{17}-3a^{16}-a^{15}+a^{14}-2a^{13}+a^{12}+5a^{11}-3a^{10}-2a^{9}+7a^{8}-a^{7}-3a^{6}+6a^{5}-5a^{4}-8a^{3}+9a^{2}+3a-1$, $2a^{28}+a^{27}-4a^{26}+3a^{25}+2a^{24}-2a^{23}+6a^{22}+a^{21}-4a^{20}+6a^{19}-5a^{17}+4a^{16}-5a^{15}-7a^{14}+6a^{13}-5a^{12}-7a^{11}+7a^{10}-5a^{9}-2a^{8}+13a^{7}-7a^{6}-2a^{5}+14a^{4}-7a^{3}+a^{2}+12a-9$, $2a^{28}+2a^{27}-2a^{25}-3a^{24}-5a^{23}-2a^{22}+a^{21}+2a^{20}+2a^{19}-a^{18}-2a^{17}-5a^{16}-5a^{15}-a^{14}+2a^{13}+4a^{12}+a^{11}+a^{10}-3a^{9}-7a^{8}-2a^{7}+5a^{5}+5a^{4}+3a^{3}+2a^{2}-6a+1$
| 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: | \( 23857799922719012 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 23857799922719012 \cdot 1}{2\cdot\sqrt{2275479017222419439026339091644104204888156487918026752}}\cr\approx \mathstrut & 2.36380763799251 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^29 + 3*x - 2)
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 + 3*x - 2, 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 + 3*x - 2);
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 + 3*x - 2);
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 | R | $28{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $26{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $17{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $28{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $29$ | $20{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | $15{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.7.0.1}{7} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $[\ ]$ |
| 2.4.4.5 | $x^{4} + 2 x + 2$ | $4$ | $1$ | $4$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
| 2.12.12.32 | $x^{12} - 4 x^{10} - 4 x^{9} + 26 x^{8} + 40 x^{7} - 4 x^{6} - 40 x^{5} + 28 x^{4} + 72 x^{3} + 24 x^{2} - 16 x + 8$ | $4$ | $3$ | $12$ | 12T129 | $[4/3, 4/3, 4/3, 4/3]_{3}^{6}$ | |
| 2.12.12.31 | $x^{12} + 2 x^{10} + 2 x^{9} + 6 x^{8} + 12 x^{7} + 32 x^{6} + 48 x^{5} + 76 x^{4} + 48 x^{3} + 40 x^{2} + 8 x + 8$ | $4$ | $3$ | $12$ | 12T205 | $[4/3, 4/3, 4/3, 4/3, 4/3, 4/3]_{3}^{6}$ | |
|
\(8447\)
| $\Q_{8447}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 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 $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
|
\(100\!\cdots\!811\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |