Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 3*x - 4)
gp: K = bnfinit(y^25 - 3*y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 3*x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 3*x - 4)
\( x^{25} - 3x - 4 \)
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: |
\(24998869940868268143413796079000415582666182623232\)
\(\medspace = 2^{48}\cdot 4813\cdot 18\!\cdots\!69\)
| 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: |
\(2\), \(4813\), \(18452\!\cdots\!94269\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{88813\!\cdots\!16697}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}-a^{5}+a^{4}-a^{3}+a^{2}-a-1$, $a^{24}-a^{23}-a^{21}+2a^{20}+a^{18}-3a^{17}+a^{16}+2a^{15}-3a^{12}+a^{11}+a^{10}+3a^{9}-4a^{8}-2a^{7}+a^{6}+2a^{5}+2a^{4}-3a^{3}-2a^{2}-2a+5$, $6a^{24}+2a^{23}-10a^{22}+9a^{21}-9a^{19}+7a^{18}+4a^{17}-13a^{16}+7a^{15}+7a^{14}-14a^{13}+8a^{12}+10a^{11}-16a^{10}+3a^{9}+15a^{8}-21a^{7}+4a^{6}+17a^{5}-21a^{4}-3a^{3}+26a^{2}-24a-17$, $a^{23}+2a^{22}-6a^{21}+5a^{20}-a^{19}+a^{18}-8a^{17}+11a^{16}-5a^{15}-a^{14}-4a^{13}+12a^{12}-8a^{11}-3a^{10}+4a^{9}+5a^{8}-9a^{7}-a^{6}+8a^{5}-4a^{4}-6a^{3}+6a^{2}+3a-9$, $2a^{24}+3a^{23}+4a^{22}+2a^{21}+2a^{19}+a^{18}-3a^{17}-3a^{16}-3a^{15}-2a^{14}+4a^{13}+5a^{12}+3a^{11}+8a^{10}+11a^{9}+7a^{8}+3a^{7}-3a^{6}-10a^{5}-5a^{4}-a^{3}-6a^{2}-a+5$, $9a^{24}-a^{23}-9a^{22}-14a^{21}-9a^{20}-3a^{19}+10a^{18}+21a^{17}+15a^{16}+2a^{15}-14a^{14}-20a^{13}-21a^{12}-14a^{11}+15a^{10}+38a^{9}+38a^{8}+7a^{7}-28a^{6}-37a^{5}-37a^{4}-21a^{3}+12a^{2}+55a+49$, $33a^{24}+32a^{23}+31a^{22}+23a^{21}+18a^{20}+6a^{19}-7a^{18}-21a^{17}-37a^{16}-54a^{15}-66a^{14}-76a^{13}-86a^{12}-85a^{11}-81a^{10}-72a^{9}-52a^{8}-21a^{7}+6a^{6}+47a^{5}+87a^{4}+126a^{3}+162a^{2}+201a+119$, $15a^{24}-a^{23}-28a^{22}+23a^{21}-20a^{20}+47a^{19}-44a^{18}+12a^{17}-16a^{16}+22a^{15}+22a^{14}-41a^{13}+11a^{12}-31a^{11}+69a^{10}-22a^{9}-4a^{8}-48a^{7}+16a^{6}+58a^{5}-5a^{4}-14a^{3}-84a^{2}+59a-19$, $a^{24}+8a^{23}+10a^{22}-2a^{21}-12a^{20}-7a^{19}-2a^{18}-3a^{17}+6a^{16}+16a^{15}+10a^{14}-4a^{13}-12a^{12}-10a^{11}-7a^{10}-11a^{9}-a^{8}+33a^{7}+39a^{6}-6a^{5}-39a^{4}-22a^{3}-3a^{2}-8a-9$, $a^{24}+a^{23}-a^{22}+4a^{21}-a^{20}+a^{19}+a^{18}+a^{17}-a^{16}+4a^{15}+a^{14}+3a^{12}+2a^{11}+6a^{9}+5a^{8}+a^{7}+6a^{6}+5a^{5}+2a^{4}+8a^{3}+10a^{2}+a+5$, $7a^{24}-3a^{23}-3a^{22}+9a^{21}-12a^{20}+17a^{19}-17a^{18}+17a^{17}-13a^{16}+10a^{15}-2a^{14}-3a^{13}+9a^{12}-16a^{11}+19a^{10}-20a^{9}+20a^{8}-18a^{7}+7a^{6}-9a^{5}-6a^{4}+11a^{3}-15a^{2}+23a-47$, $89a^{24}+32a^{23}-122a^{22}+79a^{21}+54a^{20}-140a^{19}+81a^{18}+82a^{17}-180a^{16}+78a^{15}+139a^{14}-219a^{13}+45a^{12}+197a^{11}-231a^{10}+13a^{9}+232a^{8}-250a^{7}+a^{6}+285a^{5}-292a^{4}-44a^{3}+379a^{2}-298a-419$
| 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: | \( 3640732777248095.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 3640732777248095.0 \cdot 1}{2\cdot\sqrt{24998869940868268143413796079000415582666182623232}}\cr\approx \mathstrut & 2.75668431676784 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 - 3*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 - 3*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 - 3*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 - 3*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 $S_{25}$ |
| 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 | R | $20{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.8.0.1}{8} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $[\ ]$ |
| 2.8.16.64 | $x^{8} + 2 x^{4} + 4 x + 2$ | $8$ | $1$ | $16$ | $V_4^2:(S_3\times C_2)$ | $[4/3, 4/3, 2, 7/3, 7/3]_{3}^{2}$ | |
| Deg $16$ | $8$ | $2$ | $32$ | ||||
|
\(4813\)
| $\Q_{4813}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
|
\(184\!\cdots\!269\)
| $\Q_{18\!\cdots\!69}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{18\!\cdots\!69}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ |