Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x - 1)
gp: K = bnfinit(y^25 - 5*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 5*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 5*x - 1)
\( x^{25} - 5x - 1 \)
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: | $[3, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-397484236016954131849365709987476766109466552734375\)
\(\medspace = -\,5^{25}\cdot 131\cdot 311\cdot 542960645459\cdot 60293398488673829\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(105.68\) | 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: |
\(5\), \(131\), \(311\), \(542960645459\), \(60293398488673829\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-66686\!\cdots\!53255}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: | $13$ | 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: |
$a$, $a^{12}-2$, $a^{19}-2a^{14}+2a^{9}-a^{4}-1$, $3a^{24}-a^{23}-2a^{21}-a^{20}-a^{19}+3a^{18}-a^{17}+4a^{16}-a^{15}-3a^{13}-2a^{12}-3a^{11}+5a^{10}+7a^{8}-2a^{7}-a^{6}-5a^{5}-2a^{4}-5a^{3}+6a^{2}-2$, $18a^{24}-22a^{23}+16a^{22}-5a^{21}-4a^{20}+8a^{19}-9a^{18}+11a^{17}-14a^{16}+15a^{15}-10a^{14}+11a^{12}-19a^{11}+24a^{10}-27a^{9}+27a^{8}-16a^{7}-11a^{6}+49a^{5}-77a^{4}+72a^{3}-27a^{2}-37a-4$, $16a^{24}+13a^{23}+19a^{22}+19a^{21}+13a^{20}+20a^{19}+22a^{18}+18a^{17}+29a^{16}+32a^{15}+25a^{14}+34a^{13}+34a^{12}+25a^{11}+39a^{10}+45a^{9}+39a^{8}+57a^{7}+61a^{6}+46a^{5}+61a^{4}+65a^{3}+52a^{2}+80a+15$, $3a^{24}-7a^{23}+2a^{21}-8a^{20}+10a^{19}-6a^{18}+8a^{17}+6a^{16}-2a^{15}+18a^{14}-4a^{13}+13a^{12}+3a^{11}+2a^{10}+11a^{9}-10a^{8}+9a^{7}-14a^{6}-15a^{4}-20a^{3}-37a-8$, $5a^{24}-2a^{23}+2a^{22}-5a^{21}+7a^{20}-4a^{19}-6a^{18}+10a^{17}-5a^{16}+2a^{15}-2a^{14}+6a^{13}-2a^{12}-13a^{11}+17a^{10}-9a^{9}-2a^{8}+7a^{7}+2a^{6}+3a^{5}-21a^{4}+25a^{3}-13a^{2}-13a-3$, $13a^{24}+16a^{23}+21a^{22}+23a^{21}+27a^{20}+31a^{19}+24a^{18}+10a^{17}-2a^{16}-16a^{15}-31a^{14}-34a^{13}-32a^{12}-40a^{11}-50a^{10}-54a^{9}-66a^{8}-77a^{7}-60a^{6}-25a^{5}+2a^{4}+34a^{3}+73a^{2}+86a+13$, $2a^{24}+4a^{23}+9a^{22}+6a^{21}+11a^{20}+6a^{19}+7a^{18}+6a^{17}+4a^{16}+7a^{15}+3a^{14}+2a^{13}-4a^{12}-12a^{11}-15a^{10}-24a^{9}-18a^{8}-24a^{7}-13a^{6}-15a^{5}-14a^{4}-7a^{3}-15a^{2}+3a$, $10a^{24}-3a^{23}-21a^{22}+5a^{21}-20a^{20}+10a^{19}+11a^{18}+4a^{17}+24a^{16}-10a^{15}-6a^{14}-27a^{13}-21a^{12}-4a^{11}+21a^{10}+45a^{9}+14a^{8}+19a^{7}-63a^{6}-42a^{5}-22a^{4}-21a^{3}+102a^{2}+12a-1$, $9a^{24}-4a^{23}-4a^{22}+2a^{21}+a^{20}+a^{19}-10a^{18}+7a^{17}+9a^{16}-20a^{15}+9a^{14}+14a^{13}-18a^{12}+5a^{11}+9a^{10}-a^{9}-3a^{8}-6a^{7}+23a^{6}-11a^{5}-23a^{4}+41a^{3}-16a^{2}-32a-6$, $582a^{24}-563a^{23}+208a^{22}+354a^{21}-734a^{20}+728a^{19}-222a^{18}-456a^{17}+974a^{16}-898a^{15}+288a^{14}+635a^{13}-1221a^{12}+1165a^{11}-300a^{10}-800a^{9}+1603a^{8}-1408a^{7}+369a^{6}+1138a^{5}-2028a^{4}+1855a^{3}-372a^{2}-1415a-248$
| 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: | \( 3544806216556346.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{11}\cdot 3544806216556346.0 \cdot 1}{2\cdot\sqrt{397484236016954131849365709987476766109466552734375}}\cr\approx \mathstrut & 0.428519991057905 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 - 5*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^25 - 5*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^25 - 5*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^25 - 5*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 15511210043330985984000000 |
| The 1958 conjugacy class representatives for $S_{25}$ |
| Character table for $S_{25}$ |
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 | ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/11.5.0.1}{5} }$ | $15{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | $23{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | $23{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $25$ | $25$ | $1$ | $25$ | |||
|
\(131\)
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 131.3.0.1 | $x^{3} + 3 x + 129$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 131.18.0.1 | $x^{18} - x + 126$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
|
\(311\)
| 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 $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
|
\(542960645459\)
| $\Q_{542960645459}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{542960645459}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(60293398488673829\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ |