Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^33 + 2*x - 1)
gp: K = bnfinit(y^33 + 2*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^33 + 2*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^33 + 2*x - 1)
\( x^{33} + 2x - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $33$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(12554203599883401615432606686031362766869246428265252406305\)
\(\medspace = 5\cdot 25\!\cdots\!61\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(57.62\) | 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: | $5^{1/2}2510840719976680323086521337206272553373849285653050481261^{1/2}\approx 1.1204554252572212e+29$ | ||
| Ramified primes: |
\(5\), \(25108\!\cdots\!81261\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{12554\!\cdots\!06305}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$
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: | $16$ | 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: |
$a^{32}+2$, $a^{21}-a^{10}+1$, $a^{30}+a^{28}-a^{24}-a^{22}+a^{18}+a^{16}-a^{12}-a^{10}+a^{6}+a^{4}-1$, $a^{32}+a^{31}-a^{30}+a^{28}-a^{27}+a^{25}-a^{24}+a^{22}-a^{21}+a^{19}-a^{18}+a^{16}-a^{15}+a^{13}-a^{12}+a^{10}-a^{9}+a^{7}-a^{6}+a^{4}-a^{3}+2a$, $a^{32}+a^{28}-a^{25}+a^{22}-a^{19}+a^{16}-a^{13}+a^{10}-a^{7}+a^{4}-a+2$, $a^{32}+a^{31}+a^{26}-a^{22}+a^{21}+a^{18}-2a^{17}+a^{16}-a^{14}+2a^{13}-2a^{12}+a^{10}-2a^{9}+2a^{8}-a^{7}-a^{6}+a^{5}-a^{4}+a^{3}-a+2$, $a^{32}+a^{31}+a^{30}+a^{25}-a^{24}-2a^{22}+2a^{19}+a^{17}-2a^{16}+2a^{15}-a^{14}+2a^{13}-2a^{12}-2a^{10}+a^{9}-a^{8}+2a^{7}-2a^{6}+2a^{5}-a^{4}+2a^{3}+1$, $a^{31}+a^{30}+a^{29}-a^{26}-a^{25}-a^{24}+a^{23}+a^{22}+a^{21}-a^{20}-a^{19}-a^{18}+a^{17}+a^{16}+a^{15}-a^{13}-a^{12}-a^{11}+2a^{8}+a^{7}-2a^{5}-a^{4}+2a^{2}+a$, $a^{31}-a^{30}-a^{29}+a^{28}-a^{26}+a^{25}-a^{23}+a^{22}+a^{21}-a^{20}-a^{19}+2a^{18}-2a^{16}+a^{15}+a^{14}-2a^{13}+a^{12}+a^{11}-a^{10}-a^{9}+2a^{8}-2a^{6}+a^{5}+a^{4}-3a^{3}+a^{2}+2a-2$, $2a^{32}+3a^{31}+3a^{30}+3a^{29}+3a^{28}+2a^{27}+2a^{26}+2a^{25}+a^{24}+a^{23}-a^{18}-a^{17}-a^{16}-a^{15}-a^{13}-a^{12}-a^{10}-a^{8}-2a^{7}-a^{6}-2a^{5}-2a^{4}-2a^{3}-3a^{2}-3a+2$, $2a^{32}-a^{31}-4a^{30}-2a^{29}+2a^{28}+a^{27}+2a^{25}+3a^{24}-2a^{23}-3a^{22}-a^{21}-3a^{19}+4a^{17}+3a^{16}-2a^{15}-3a^{12}-5a^{11}+2a^{10}+4a^{9}-a^{7}+4a^{6}+a^{5}-5a^{4}-4a^{3}+2a^{2}+1$, $3a^{32}-a^{31}-2a^{30}+a^{29}-a^{27}+4a^{25}-2a^{24}-a^{23}+3a^{22}-3a^{21}-a^{20}+2a^{18}-a^{17}+4a^{15}-3a^{14}-a^{13}-3a^{10}+2a^{9}+5a^{8}-5a^{7}+4a^{6}+a^{5}-4a^{4}-a^{3}+2a+1$, $a^{32}+2a^{31}+2a^{30}+a^{29}+a^{28}+a^{26}+a^{24}-a^{23}-2a^{21}-a^{20}-2a^{19}-a^{18}-a^{17}+a^{11}+2a^{10}+2a^{9}+2a^{8}-a^{5}+a^{4}+a^{2}-2a$, $a^{32}-2a^{31}+a^{30}-3a^{29}+2a^{28}-a^{27}+2a^{26}-2a^{25}+2a^{24}+2a^{22}-a^{21}+a^{20}+a^{19}+a^{18}-a^{17}-a^{16}-a^{13}-2a^{12}-2a^{8}+a^{7}+3a^{5}-3a^{4}+3a^{3}-2a^{2}+5a-3$, $a^{32}+3a^{31}-a^{30}-2a^{29}+2a^{27}-a^{26}-2a^{25}+2a^{24}+a^{23}-3a^{22}-a^{21}+2a^{20}+a^{19}-2a^{18}+4a^{16}-2a^{15}-3a^{14}+2a^{13}+3a^{12}-a^{11}-3a^{10}+3a^{9}+2a^{8}-5a^{7}-a^{6}+4a^{5}-3a^{3}-a^{2}+5a-1$, $a^{31}-a^{30}-a^{29}+a^{28}+2a^{25}-2a^{23}+2a^{21}-a^{20}+a^{18}-2a^{17}-a^{16}+2a^{15}+a^{14}-2a^{13}-2a^{10}+a^{9}+2a^{8}-a^{7}-2a^{6}+a^{5}+2a^{4}-a^{3}+2a^{2}+a-2$
| 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: | \( 12254017574219786 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{16}\cdot 12254017574219786 \cdot 1}{2\cdot\sqrt{12554203599883401615432606686031362766869246428265252406305}}\cr\approx \mathstrut & 0.645300213457565 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^33 + 2*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^33 + 2*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^33 + 2*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^33 + 2*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 8683317618811886495518194401280000000 |
| The 10143 conjugacy class representatives for $S_{33}$ are not computed |
| Character table for $S_{33}$ 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 | ${\href{/padicField/2.10.0.1}{10} }^{3}{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/3.7.0.1}{7} }$ | R | $33$ | $32{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | $18{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/23.14.0.1}{14} }$ | $18{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $24{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.7.0.1 | $x^{7} + 3 x + 3$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 5.22.0.1 | $x^{22} + x^{12} + 3 x^{11} + 4 x^{9} + 3 x^{8} + 2 x^{6} + 2 x^{5} + 4 x^{3} + 3 x^{2} + 3 x + 2$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | |
|
\(251\!\cdots\!261\)
| $\Q_{25\!\cdots\!61}$ | $x$ | $1$ | $1$ | $0$ | 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 $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |