Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^29 + 3*x - 1)
gp: K = bnfinit(y^29 + 3*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 + 3*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 + 3*x - 1)
\( x^{29} + 3x - 1 \)
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: |
\(2274789759221598409627081921061651046871631560367033357\)
\(\medspace = 17\cdot 41\cdot 74959\cdot 160217\cdot 2662789\cdot 25396823\cdot 76318073627\cdot 52654144463043283\)
| 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: | $17^{1/2}41^{1/2}74959^{1/2}160217^{1/2}2662789^{1/2}25396823^{1/2}76318073627^{1/2}52654144463043283^{1/2}\approx 1.508240617150194e+27$ | ||
| Ramified primes: |
\(17\), \(41\), \(74959\), \(160217\), \(2662789\), \(25396823\), \(76318073627\), \(52654144463043283\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{22747\!\cdots\!33357}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a$, $a^{15}+2a^{8}+2a$, $a^{28}-a^{27}+a^{26}-a^{24}+a^{23}-a^{21}+a^{20}-a^{18}+a^{17}-a^{15}+a^{14}-a^{12}+a^{11}-a^{9}+a^{8}-a^{6}+a^{5}-a^{3}+a^{2}+1$, $a^{25}+a^{24}+a^{23}+a^{22}+a^{19}-2a^{17}-a^{16}-a^{14}-2a^{13}-a^{12}+a^{9}+3a^{6}+2a^{5}+a^{3}+2a^{2}-2$, $a^{27}-a^{23}+a^{20}+2a^{19}-2a^{18}-2a^{17}+a^{15}+3a^{14}-3a^{12}-a^{11}-a^{10}+3a^{9}+3a^{8}-2a^{7}-2a^{6}-2a^{5}+a^{4}+4a^{3}-2a^{2}+1$, $a^{28}-2a^{27}+a^{26}+a^{25}-3a^{20}+2a^{19}+a^{18}-3a^{17}+6a^{16}-2a^{15}-5a^{14}+7a^{13}-7a^{12}-a^{11}+10a^{10}-9a^{9}+4a^{8}+6a^{7}-13a^{6}+7a^{5}+a^{4}-9a^{3}+11a^{2}-2a-1$, $a^{28}+a^{27}+2a^{26}+a^{23}-a^{22}+a^{20}-a^{19}+a^{18}+2a^{17}-a^{16}-3a^{13}-a^{12}-a^{11}-3a^{10}+a^{9}+2a^{8}+4a^{6}+4a^{5}+2a^{3}-a^{2}-5a+3$, $a^{28}-2a^{27}+2a^{25}-2a^{24}+3a^{22}-3a^{21}+a^{20}+3a^{19}-4a^{18}+2a^{17}+2a^{16}-4a^{15}+2a^{14}+2a^{13}-5a^{12}+2a^{11}+3a^{10}-7a^{9}+5a^{8}+2a^{7}-8a^{6}+8a^{5}-a^{4}-6a^{3}+9a^{2}-3a-1$, $a^{28}+a^{27}+2a^{26}-2a^{24}-a^{23}+2a^{21}+3a^{20}-3a^{18}-2a^{17}+a^{16}+3a^{15}+3a^{14}-4a^{12}-2a^{11}+2a^{10}+2a^{9}+3a^{8}-4a^{6}-a^{5}+a^{3}+4a^{2}+1$, $2a^{28}+2a^{27}+2a^{26}+a^{25}+a^{23}-a^{22}-2a^{21}-a^{20}-2a^{19}-4a^{18}-3a^{17}-3a^{16}-6a^{15}-5a^{14}-5a^{13}-6a^{12}-7a^{11}-5a^{10}-6a^{9}-8a^{8}-5a^{7}-6a^{6}-8a^{5}-6a^{4}-3a^{3}-7a^{2}-5a+4$, $2a^{28}-2a^{27}+a^{26}-a^{24}+a^{23}-a^{22}+2a^{21}-2a^{20}+a^{19}-a^{18}+a^{17}-a^{14}+2a^{13}-3a^{12}+4a^{11}-6a^{10}+9a^{9}-11a^{8}+13a^{7}-16a^{6}+17a^{5}-16a^{4}+16a^{3}-16a^{2}+14a-5$, $6a^{28}-7a^{27}+13a^{25}-9a^{24}-3a^{23}+12a^{22}-3a^{21}-11a^{20}+9a^{19}+6a^{18}-20a^{17}+9a^{16}+6a^{15}-14a^{14}-8a^{13}+18a^{12}-14a^{11}-17a^{10}+19a^{9}-8a^{8}-21a^{7}+7a^{6}+15a^{5}-35a^{4}+4a^{3}+21a^{2}-28a+6$, $3a^{28}+a^{27}+2a^{26}+2a^{25}-2a^{24}-a^{23}-4a^{21}-4a^{20}-2a^{18}-4a^{17}+2a^{16}-a^{15}-2a^{14}+a^{13}+5a^{12}+3a^{10}+7a^{9}+2a^{8}+6a^{6}+a^{5}-7a^{4}-4a^{2}-11a+5$, $4a^{28}+3a^{27}-8a^{26}-a^{25}+4a^{24}-3a^{23}-2a^{22}+6a^{21}-6a^{19}+2a^{18}+2a^{17}+a^{16}+3a^{15}+3a^{14}-6a^{13}+5a^{12}+2a^{11}-a^{10}+4a^{9}+13a^{8}-12a^{7}-3a^{6}+13a^{5}+6a^{4}-14a^{3}+16a^{2}+8a-5$
| 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: | \( 6546753152959768.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 6546753152959768.0 \cdot 1}{2\cdot\sqrt{2274789759221598409627081921061651046871631560367033357}}\cr\approx \mathstrut & 0.648744204035480 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^29 + 3*x - 1)
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 - 1, 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 - 1);
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 - 1);
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 $S_{29}$ are not computed |
| Character table for $S_{29}$ is not computed |
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 | $27{,}\,{\href{/padicField/2.2.0.1}{2} }$ | $28{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $29$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | $20{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $28{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.7.0.1}{7} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.13.0.1}{13} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }$ | R | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.7.0.1}{7} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $25{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.23.0.1 | $x^{23} + 15 x^{2} + 16 x + 14$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
|
\(41\)
| 41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 41.13.0.1 | $x^{13} + 13 x + 35$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
| 41.14.0.1 | $x^{14} + 12 x^{7} + 15 x^{6} + 4 x^{5} + 27 x^{4} + 11 x^{3} + 39 x^{2} + 10 x + 6$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(74959\)
| $\Q_{74959}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{74959}$ | $x$ | $1$ | $1$ | $0$ | 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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(160217\)
| $\Q_{160217}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
|
\(2662789\)
| $\Q_{2662789}$ | $x$ | $1$ | $1$ | $0$ | 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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(25396823\)
| $\Q_{25396823}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{25396823}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{25396823}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{25396823}$ | $x$ | $1$ | $1$ | $0$ | 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(76318073627\)
| $\Q_{76318073627}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{76318073627}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(52654144463043283\)
| $\Q_{52654144463043283}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ |