Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 + x - 4)
gp: K = bnfinit(y^25 + y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 + x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 + x - 4)
\( x^{25} + x - 4 \)
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: |
\(25000000000000001333735776850284124449081472843776\)
\(\medspace = 2^{48}\cdot 13\cdot 5969653\cdot 175300357\cdot 544585241\cdot 11988341597\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(94.61\) | 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: |
\(2\), \(13\), \(5969653\), \(175300357\), \(544585241\), \(11988341597\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{88817\!\cdots\!82521}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$4a^{24}-5a^{22}-4a^{21}+a^{20}+6a^{19}+4a^{18}-3a^{17}-5a^{16}-3a^{15}+3a^{14}+5a^{13}-a^{11}-2a^{10}-a^{9}-3a^{7}+2a^{6}+6a^{5}+4a^{4}-3a^{3}-12a^{2}-5a+13$, $2a^{24}-4a^{23}-6a^{22}-4a^{21}-a^{20}+a^{19}+7a^{18}+9a^{17}-a^{16}-a^{15}-4a^{14}-13a^{13}-3a^{12}+3a^{11}+6a^{10}+9a^{9}+10a^{8}-12a^{6}-8a^{5}-10a^{4}-9a^{3}+17a^{2}+10a+13$, $5a^{24}+4a^{23}-a^{22}-5a^{21}-5a^{20}+a^{19}+7a^{18}+3a^{17}-7a^{16}-6a^{15}+7a^{14}+9a^{13}-4a^{12}-10a^{11}-4a^{10}+5a^{9}+11a^{8}+6a^{7}-8a^{6}-14a^{5}+2a^{4}+18a^{3}+2a^{2}-20a-3$, $4a^{24}+3a^{23}-4a^{22}-13a^{21}-15a^{20}-5a^{19}+6a^{18}+8a^{17}+2a^{16}-12a^{15}-21a^{14}-12a^{13}+2a^{12}+8a^{11}+4a^{10}-14a^{9}-29a^{8}-21a^{7}-a^{6}+15a^{5}+15a^{4}-6a^{3}-31a^{2}-35a-9$, $a^{24}+7a^{23}-a^{22}-10a^{21}+5a^{20}+11a^{19}-11a^{18}-9a^{17}+18a^{16}+3a^{15}-24a^{14}+8a^{13}+26a^{12}-24a^{11}-16a^{10}+34a^{9}-a^{8}-32a^{7}+13a^{6}+28a^{5}-25a^{4}-18a^{3}+32a^{2}+5a-27$, $7a^{24}-6a^{23}+8a^{22}-4a^{20}+15a^{19}-13a^{18}+10a^{17}+7a^{16}-14a^{15}+22a^{14}-12a^{13}+4a^{12}+15a^{11}-18a^{10}+22a^{9}-7a^{8}-a^{7}+16a^{6}-11a^{5}+17a^{4}-10a^{3}+13a^{2}+4a-3$, $8a^{24}-19a^{23}+13a^{22}+5a^{21}-22a^{20}+23a^{19}-a^{18}-31a^{17}+44a^{16}-22a^{15}-21a^{14}+56a^{13}-61a^{12}+26a^{11}+38a^{10}-89a^{9}+80a^{8}-11a^{7}-66a^{6}+97a^{5}-67a^{4}-5a^{3}+76a^{2}-93a+43$, $14a^{24}-a^{23}-28a^{22}-14a^{21}+25a^{20}+34a^{19}+17a^{18}-30a^{17}+2a^{16}+22a^{15}+71a^{14}-4a^{13}-10a^{12}-49a^{11}+57a^{10}+23a^{9}+33a^{8}-98a^{7}-25a^{6}-22a^{5}+82a^{4}-44a^{3}-63a^{2}-112a+53$, $7a^{24}+7a^{23}+2a^{22}+4a^{21}+14a^{20}+11a^{19}-2a^{18}+4a^{17}+14a^{16}+3a^{15}-4a^{14}-3a^{13}-a^{12}+2a^{11}-9a^{10}-19a^{9}-8a^{8}+7a^{7}-14a^{6}-27a^{5}+5a^{4}+9a^{3}-7a^{2}-5a+13$, $a^{24}+3a^{23}-5a^{22}+4a^{21}+2a^{20}-6a^{19}+5a^{18}-2a^{17}-5a^{16}+8a^{15}-5a^{14}+8a^{12}-10a^{11}+3a^{10}+6a^{9}-14a^{8}+10a^{7}+2a^{6}-11a^{5}+14a^{4}-5a^{3}-9a^{2}+18a-17$, $37a^{24}-9a^{23}-57a^{22}+73a^{21}-96a^{20}+53a^{19}-4a^{18}-69a^{17}+110a^{16}-119a^{15}+80a^{14}+14a^{13}-82a^{12}+170a^{11}-150a^{10}+112a^{9}+25a^{8}-120a^{7}+233a^{6}-212a^{5}+138a^{4}+17a^{3}-195a^{2}+289a-271$, $24a^{24}-50a^{23}-58a^{22}+15a^{21}+70a^{20}+22a^{19}-71a^{18}-80a^{17}+23a^{16}+105a^{15}+52a^{14}-74a^{13}-97a^{12}+25a^{11}+133a^{10}+62a^{9}-109a^{8}-155a^{7}+3a^{6}+159a^{5}+94a^{4}-117a^{3}-177a^{2}+23a+253$
| 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: | \( 2845294424971493.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 2845294424971493.0 \cdot 1}{2\cdot\sqrt{25000000000000001333735776850284124449081472843776}}\cr\approx \mathstrut & 2.15434686411688 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 + x - 4)
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 + x - 4, 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 + x - 4);
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 + x - 4);
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 | $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.4.0.1}{4} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | $15{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | $16{,}\,{\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $24{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $15{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | $22{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | $22{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.8.16.63 | $x^{8} + 2 x^{4} + 4 x + 6$ | $8$ | $1$ | $16$ | $V_4^2:(S_3\times C_2)$ | $[4/3, 4/3, 2, 7/3, 7/3]_{3}^{2}$ | |
| Deg $16$ | $8$ | $2$ | $32$ | ||||
|
\(13\)
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.5.0.1 | $x^{5} + 4 x + 11$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 13.15.0.1 | $x^{15} + 2 x^{7} + 12 x^{6} + 2 x^{5} + 11 x^{4} + 10 x^{3} + 11 x^{2} + 8 x + 11$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
|
\(5969653\)
| $\Q_{5969653}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{5969653}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{5969653}$ | $x$ | $1$ | $1$ | $0$ | 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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(175300357\)
| 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 $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(544585241\)
| $\Q_{544585241}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
|
\(11988341597\)
| $\Q_{11988341597}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |