Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 + 3*x - 5)
gp: K = bnfinit(y^25 + 3*y - 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 + 3*x - 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 + 3*x - 5)
\( x^{25} + 3x - 5 \)
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: |
\(5293957050398508851033601833168761847238531327142393\)
\(\medspace = 7\cdot 79\cdot 367\cdot 26\!\cdots\!43\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(117.21\) | 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: | $7^{1/2}79^{1/2}367^{1/2}26084902515378139802383835670525209766093940543^{1/2}\approx 7.275958390754107e+25$ | ||
| Ramified primes: |
\(7\), \(79\), \(367\), \(26084\!\cdots\!40543\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{52939\!\cdots\!42393}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a-1$, $a^{24}-4a^{23}-a^{22}+2a^{21}-4a^{20}+a^{19}-7a^{18}+3a^{16}-5a^{15}+6a^{14}-10a^{13}+2a^{12}+9a^{11}-8a^{10}+14a^{9}-11a^{8}+18a^{6}-13a^{5}+18a^{4}-11a^{3}-7a^{2}+23a-14$, $8a^{24}+9a^{23}+5a^{22}-4a^{21}-8a^{20}-8a^{19}-12a^{18}-14a^{17}-6a^{16}+2a^{15}+2a^{14}+8a^{13}+19a^{12}+18a^{11}+7a^{10}+7a^{9}+4a^{8}-13a^{7}-25a^{6}-15a^{5}-13a^{4}-19a^{3}-8a^{2}+17a+41$, $27a^{24}-23a^{23}+2a^{22}+17a^{21}-24a^{19}+33a^{18}+4a^{17}-44a^{16}+53a^{15}-11a^{14}-25a^{13}+12a^{12}+39a^{11}-65a^{10}+11a^{9}+72a^{8}-107a^{7}+41a^{6}+34a^{5}-39a^{4}-58a^{3}+121a^{2}-54a-31$, $37a^{24}-133a^{23}-142a^{22}+39a^{21}+213a^{20}+96a^{19}-186a^{18}-225a^{17}+49a^{16}+293a^{15}+132a^{14}-219a^{13}-289a^{12}-4a^{11}+355a^{10}+268a^{9}-229a^{8}-472a^{7}-85a^{6}+534a^{5}+416a^{4}-324a^{3}-623a^{2}-134a+724$, $19a^{24}+16a^{23}+3a^{22}-24a^{21}-18a^{20}+11a^{19}+21a^{18}+16a^{17}-24a^{16}-13a^{15}-5a^{14}+23a^{13}+7a^{12}-18a^{11}-7a^{10}+a^{9}+40a^{8}-18a^{7}-26a^{6}-35a^{5}+37a^{4}+66a^{3}-3a^{2}-45a-41$, $a^{24}+14a^{23}+3a^{21}-7a^{20}-15a^{19}+10a^{18}+19a^{16}+2a^{15}-15a^{14}+a^{13}-25a^{12}+25a^{11}+13a^{10}+a^{9}+15a^{8}-40a^{7}-a^{6}+a^{5}+12a^{4}+42a^{3}-21a^{2}-2a-24$, $3a^{24}-3a^{23}-4a^{22}+a^{21}-5a^{20}-a^{19}-3a^{18}-a^{17}+4a^{16}-6a^{15}+7a^{14}+5a^{13}-4a^{12}+4a^{11}-5a^{10}+a^{9}-8a^{8}-7a^{7}+10a^{6}-10a^{5}+4a^{4}+7a^{3}+6a^{2}+13a-1$, $4a^{24}+13a^{23}-2a^{22}-17a^{21}-7a^{20}+19a^{19}+30a^{18}+13a^{17}-10a^{16}-10a^{15}+11a^{14}+11a^{13}-24a^{12}-49a^{11}-25a^{10}+25a^{9}+48a^{8}+8a^{7}-47a^{6}-26a^{5}+57a^{4}+92a^{3}+31a^{2}-73a-94$, $4a^{24}+4a^{23}-5a^{22}-7a^{21}+3a^{20}+8a^{19}-3a^{18}-11a^{17}+3a^{16}+12a^{15}-3a^{14}-12a^{13}+13a^{11}+4a^{10}-15a^{9}-7a^{8}+17a^{7}+16a^{6}-17a^{5}-20a^{4}+23a^{3}+24a^{2}-17a-16$, $10a^{24}+12a^{23}-6a^{22}-16a^{21}+4a^{20}+21a^{19}+2a^{18}-27a^{17}-4a^{16}+30a^{15}+8a^{14}-32a^{13}-11a^{12}+31a^{11}+18a^{10}-28a^{9}-32a^{8}+27a^{7}+47a^{6}-23a^{5}-65a^{4}+26a^{3}+71a^{2}-16a-51$, $1492a^{24}+1224a^{23}+726a^{22}+307a^{21}-16a^{20}-452a^{19}-1088a^{18}-1724a^{17}-1999a^{16}-1623a^{15}-549a^{14}+902a^{13}+2080a^{12}+2424a^{11}+1939a^{10}+1171a^{9}+595a^{8}+156a^{7}-520a^{6}-1607a^{5}-2837a^{4}-3589a^{3}-3118a^{2}-1105a+6209$
| 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: | \( 11130177971379280 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 11130177971379280 \cdot 1}{2\cdot\sqrt{5293957050398508851033601833168761847238531327142393}}\cr\approx \mathstrut & 0.579122345109288 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 + 3*x - 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 + 3*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 + 3*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 + 3*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 | ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | $20{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.8.0.1}{8} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | $23{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | $24{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.23.0.1 | $x^{23} + 4 x^{2} + 4 x + 4$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
|
\(79\)
| $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.9.0.1 | $x^{9} + 57 x^{2} + 19 x + 76$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| 79.10.0.1 | $x^{10} + 4 x^{6} + 44 x^{5} + 51 x^{4} + x^{3} + 30 x^{2} + 42 x + 3$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(367\)
| $\Q_{367}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
|
\(260\!\cdots\!543\)
| $\Q_{26\!\cdots\!43}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{26\!\cdots\!43}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |