Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 + 5*x - 5)
gp: K = bnfinit(y^25 + 5*y - 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 + 5*x - 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 + 5*x - 5)
\( x^{25} + 5x - 5 \)
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: |
\(5691440156356331339844223309247762262821197509765625\)
\(\medspace = 5^{25}\cdot 127\cdot 390538849\cdot 963885163\cdot 399465067602449\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(117.55\) | 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: |
\(5\), \(127\), \(390538849\), \(963885163\), \(399465067602449\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{95486\!\cdots\!84505}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a-1$, $a^{24}-a^{23}+3a^{22}-3a^{21}+4a^{20}-4a^{19}+2a^{18}-2a^{17}-2a^{16}+2a^{15}-6a^{14}+6a^{13}-8a^{12}+8a^{11}-6a^{10}+4a^{9}+a^{8}-5a^{7}+11a^{6}-16a^{5}+16a^{4}-17a^{3}+12a^{2}-10a+4$, $a^{23}+a^{22}+4a^{20}-2a^{19}+5a^{18}+3a^{16}+3a^{15}+5a^{13}+6a^{11}-4a^{10}+9a^{9}-4a^{8}+4a^{7}+4a^{6}-9a^{5}+14a^{4}-11a^{3}+8a^{2}-8a+6$, $a^{24}-4a^{23}+6a^{22}-3a^{21}-4a^{20}+9a^{19}-3a^{18}-2a^{17}+6a^{16}-7a^{15}+2a^{14}+5a^{13}-9a^{12}+5a^{11}-3a^{10}-5a^{9}+17a^{8}-13a^{7}-4a^{6}+17a^{5}-26a^{4}+11a^{3}+24a^{2}-44a+26$, $74a^{24}+78a^{23}+66a^{22}+66a^{21}+50a^{20}+43a^{19}+26a^{18}+16a^{17}-2a^{16}-16a^{15}-26a^{14}-46a^{13}-49a^{12}-67a^{11}-61a^{10}-86a^{9}-59a^{8}-90a^{7}-54a^{6}-81a^{5}-31a^{4}-73a^{3}-47a+394$, $12a^{24}+22a^{23}+24a^{22}+20a^{21}+11a^{20}-5a^{19}-15a^{18}-15a^{17}-3a^{16}+10a^{15}+24a^{14}+35a^{13}+25a^{12}+5a^{11}-15a^{10}-25a^{9}-29a^{8}-16a^{7}+16a^{6}+42a^{5}+42a^{4}+29a^{3}+13a^{2}-28a+6$, $14a^{24}+37a^{23}+11a^{22}-35a^{21}-39a^{20}+11a^{19}+53a^{18}+27a^{17}-41a^{16}-62a^{15}+a^{14}+73a^{13}+54a^{12}-44a^{11}-95a^{10}-20a^{9}+94a^{8}+92a^{7}-40a^{6}-135a^{5}-55a^{4}+114a^{3}+149a^{2}-20a-116$, $131a^{24}+160a^{23}+122a^{22}+82a^{21}+15a^{20}-75a^{19}-115a^{18}-199a^{17}-166a^{16}-186a^{15}-88a^{14}-24a^{13}+70a^{12}+184a^{11}+187a^{10}+284a^{9}+160a^{8}+182a^{7}-7a^{6}-75a^{5}-190a^{4}-315a^{3}-232a^{2}-357a+584$, $38a^{24}-2a^{23}-29a^{22}+41a^{21}+8a^{20}-47a^{19}+39a^{18}+29a^{17}-64a^{16}+26a^{15}+55a^{14}-84a^{13}+90a^{11}-94a^{10}-37a^{9}+131a^{8}-82a^{7}-90a^{6}+165a^{5}-49a^{4}-170a^{3}+191a^{2}+26a-69$, $157a^{24}-159a^{23}-299a^{22}-40a^{21}+314a^{20}+280a^{19}-153a^{18}-436a^{17}-148a^{16}+392a^{15}+462a^{14}-109a^{13}-614a^{12}-331a^{11}+463a^{10}+728a^{9}+9a^{8}-835a^{7}-635a^{6}+491a^{5}+1099a^{4}+256a^{3}-1075a^{2}-1102a+1201$, $4a^{24}-81a^{23}+147a^{22}-153a^{21}+100a^{20}+14a^{19}-137a^{18}+230a^{17}-231a^{16}+133a^{15}+50a^{14}-244a^{13}+376a^{12}-363a^{11}+192a^{10}+100a^{9}-401a^{8}+588a^{7}-543a^{6}+248a^{5}+221a^{4}-686a^{3}+945a^{2}-830a+346$, $20a^{24}-36a^{23}+33a^{22}-10a^{21}+25a^{20}+25a^{19}-17a^{18}+49a^{17}-65a^{16}+62a^{15}-66a^{14}+58a^{13}-5a^{12}+10a^{11}+72a^{10}-88a^{9}+118a^{8}-167a^{7}+119a^{6}-132a^{5}+61a^{4}+24a^{3}-77a^{2}+199a-161$
| 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: | \( 7950768287499577.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 7950768287499577.0 \cdot 1}{2\cdot\sqrt{5691440156356331339844223309247762262821197509765625}}\cr\approx \mathstrut & 0.398984872967519 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 + 5*x - 5)
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 - 5, 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 - 5);
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 - 5);
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 | ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | R | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | $17{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/53.6.0.1}{6} }$ | $25$ |
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 $25$ | $25$ | $1$ | $25$ | |||
|
\(127\)
| 127.2.0.1 | $x^{2} + 126 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 127.2.1.1 | $x^{2} + 381$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.0.1 | $x^{2} + 126 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 127.19.0.1 | $x^{19} + 30 x + 124$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | |
|
\(390538849\)
| 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
|
\(963885163\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(399465067602449\)
| $\Q_{399465067602449}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |