Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 + 4*x - 2)
gp: K = bnfinit(y^25 + 4*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 + 4*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 + 4*x - 2)
\( x^{25} + 4x - 2 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1501654477024529031788831942140931778058569908224\)
\(\medspace = 2^{24}\cdot 29\cdot 11136449\cdot 3349154609\cdot 82750417248561907056701\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(84.54\) | 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: | $2^{24/25}29^{1/2}11136449^{1/2}3349154609^{1/2}82750417248561907056701^{1/2}\approx 5.819877660964827e+20$ | ||
| Ramified primes: |
\(2\), \(29\), \(11136449\), \(3349154609\), \(82750417248561907056701\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{89505\!\cdots\!96089}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$
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: | $12$ | 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: |
$2a^{19}-4a^{13}+4a^{7}-1$, $a^{19}-a^{13}+2a-1$, $7a^{24}+6a^{23}+2a^{22}-4a^{21}-9a^{20}-10a^{19}-7a^{18}-a^{17}+6a^{16}+11a^{15}+11a^{14}+5a^{13}-3a^{12}-10a^{11}-14a^{10}-12a^{9}-4a^{8}+7a^{7}+15a^{6}+17a^{5}+13a^{4}+2a^{3}-11a^{2}-19a+9$, $2a^{24}+3a^{23}+3a^{22}+a^{21}+2a^{20}+3a^{19}+a^{18}+a^{17}+3a^{16}+a^{15}-a^{14}+2a^{13}-4a^{11}-a^{9}-7a^{8}-2a^{7}-a^{6}-9a^{5}-3a^{4}-9a^{2}-4a+9$, $3a^{24}+2a^{23}-3a^{22}+5a^{21}-5a^{20}+a^{19}-7a^{17}+6a^{16}-8a^{15}+2a^{14}+4a^{13}-8a^{12}+13a^{11}-6a^{10}+5a^{9}+8a^{8}-10a^{7}+16a^{6}-12a^{5}+2a^{4}+7a^{3}-22a^{2}+19a-5$, $6a^{24}+5a^{23}+5a^{22}+2a^{21}+3a^{20}+3a^{19}+2a^{18}+6a^{17}+5a^{16}+2a^{15}+4a^{14}+a^{13}+2a^{12}+7a^{11}+2a^{10}+2a^{9}+3a^{8}-3a^{7}+2a^{6}+3a^{5}-4a^{4}+2a^{3}-a^{2}-7a+25$, $a^{24}-a^{23}-3a^{20}-2a^{19}-a^{18}+3a^{17}+3a^{16}+a^{15}-a^{14}+a^{13}+2a^{12}-6a^{11}-4a^{10}+4a^{8}+a^{7}+a^{6}+3a^{5}+2a^{4}+3a^{3}-10a^{2}-7a+5$, $6a^{24}+a^{23}+4a^{21}+2a^{20}-2a^{19}+2a^{18}+4a^{17}-2a^{16}-2a^{15}+3a^{14}-4a^{12}+2a^{10}-a^{9}-a^{8}+a^{7}+4a^{6}+2a^{5}-2a^{4}+2a^{3}+8a^{2}-4a+17$, $a^{24}+2a^{23}+5a^{22}+2a^{21}-2a^{20}+4a^{18}-6a^{16}-a^{15}+4a^{14}-a^{12}+2a^{11}+2a^{10}+a^{9}-a^{8}-3a^{7}-3a^{6}-a^{5}+5a^{4}+2a^{3}-7a^{2}+4a+17$, $13a^{24}-12a^{23}+9a^{22}-5a^{21}-a^{20}+7a^{19}-12a^{18}+17a^{17}-18a^{16}+18a^{15}-13a^{14}+7a^{13}+2a^{12}-12a^{11}+20a^{10}-27a^{9}+28a^{8}-26a^{7}+16a^{6}-2a^{5}-16a^{4}+36a^{3}-54a^{2}+70a-23$, $4a^{24}+a^{23}+3a^{22}+4a^{21}+5a^{20}-2a^{19}+4a^{18}-3a^{17}+7a^{16}-2a^{15}-7a^{14}+4a^{13}-5a^{12}+2a^{11}-11a^{10}-3a^{9}-6a^{8}+2a^{7}-13a^{6}-14a^{5}+7a^{4}-18a^{3}+6a^{2}-24a+13$, $2a^{24}+15a^{23}-2a^{22}-20a^{21}-5a^{20}+19a^{19}+9a^{18}-20a^{17}-13a^{16}+21a^{15}+24a^{14}-16a^{13}-27a^{12}+12a^{11}+37a^{10}-5a^{9}-39a^{8}-5a^{7}+46a^{6}+14a^{5}-45a^{4}-26a^{3}+52a^{2}+46a-31$
| 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: | \( 281396393824665.03 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 281396393824665.03 \cdot 1}{2\cdot\sqrt{1501654477024529031788831942140931778058569908224}}\cr\approx \mathstrut & 0.869344558749160 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 + 4*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^25 + 4*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^25 + 4*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^25 + 4*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 15511210043330985984000000 |
| The 1958 conjugacy class representatives for $S_{25}$ are not computed |
| Character table for $S_{25}$ 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 | R | $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }$ | $16{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $25$ | R | $20{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $25$ | $25$ | $1$ | $24$ | |||
|
\(29\)
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.22.0.1 | $x^{22} - x + 2$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | |
|
\(11136449\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
|
\(3349154609\)
| $\Q_{3349154609}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
|
\(827\!\cdots\!701\)
| $\Q_{82\!\cdots\!01}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ |