Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 + 2*x - 5)
gp: K = bnfinit(y^25 + 2*y - 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 + 2*x - 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 + 2*x - 5)
\( x^{25} + 2x - 5 \)
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: |
\(5293955920384129865607305662082268504564193838180857\)
\(\medspace = 137\cdot 3187\cdot 319234466347\cdot 37\!\cdots\!49\)
| 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: | $137^{1/2}3187^{1/2}319234466347^{1/2}37981133025939497114459508407014049^{1/2}\approx 7.27595761421418e+25$ | ||
| Ramified primes: |
\(137\), \(3187\), \(319234466347\), \(37981\!\cdots\!14049\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{52939\!\cdots\!80857}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$2a^{22}-a^{21}+2a^{17}-a^{16}-2a^{14}+3a^{13}-a^{12}-a^{11}+a^{10}-3a^{9}+2a^{8}-2a^{7}+4a^{6}-6a^{5}+3a^{4}-2a^{3}+5a-4$, $a^{24}-a^{23}+4a^{22}-4a^{21}+5a^{20}-5a^{19}+4a^{18}-2a^{17}+a^{16}+3a^{15}-4a^{14}+6a^{13}-8a^{12}+5a^{11}-6a^{10}+3a^{8}-9a^{7}+14a^{6}-19a^{5}+20a^{4}-23a^{3}+17a^{2}-14a+7$, $18a^{24}-25a^{23}+28a^{22}-34a^{21}+40a^{20}-43a^{19}+51a^{18}-54a^{17}+58a^{16}-64a^{15}+63a^{14}-67a^{13}+67a^{12}-63a^{11}+64a^{10}-55a^{9}+49a^{8}-41a^{7}+23a^{6}-13a^{5}-7a^{4}+30a^{3}-49a^{2}+83a-72$, $a^{24}-a^{23}+a^{22}-a^{21}+a^{20}-a^{19}+a^{18}-a^{17}+a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}-a^{5}+a^{4}-a^{3}+3a-4$, $4a^{24}-7a^{23}-15a^{22}-15a^{21}-6a^{20}+8a^{19}+20a^{18}+23a^{17}+14a^{16}-3a^{15}-19a^{14}-25a^{13}-17a^{12}+2a^{11}+23a^{10}+34a^{9}+29a^{8}+9a^{7}-17a^{6}-36a^{5}-36a^{4}-15a^{3}+18a^{2}+46a+62$, $19a^{24}+23a^{23}+20a^{22}+19a^{21}+17a^{20}+9a^{19}+9a^{18}-5a^{17}-4a^{16}-18a^{15}-20a^{14}-28a^{13}-35a^{12}-32a^{11}-42a^{10}-28a^{9}-39a^{8}-19a^{7}-22a^{6}-5a^{5}+9a^{4}+11a^{3}+41a^{2}+29a+104$, $6a^{23}+3a^{22}-a^{21}-a^{20}-5a^{19}+11a^{17}-2a^{16}-12a^{15}+2a^{14}-3a^{12}+8a^{11}-9a^{10}-16a^{9}+12a^{8}+8a^{7}-9a^{6}+a^{5}-3a^{4}-5a^{3}+18a^{2}+13a-19$, $2a^{24}-15a^{23}-19a^{22}-25a^{21}-47a^{20}-86a^{19}-124a^{18}-131a^{17}-82a^{16}+26a^{15}+164a^{14}+277a^{13}+306a^{12}+218a^{11}+28a^{10}-202a^{9}-384a^{8}-441a^{7}-344a^{6}-127a^{5}+126a^{4}+319a^{3}+386a^{2}+321a+183$, $29a^{24}+17a^{23}-39a^{22}+a^{21}+55a^{20}-8a^{19}-45a^{18}+41a^{17}+49a^{16}-60a^{15}-40a^{14}+64a^{13}-10a^{12}-108a^{11}+18a^{10}+100a^{9}-53a^{8}-68a^{7}+128a^{6}+75a^{5}-142a^{4}-21a^{3}+161a^{2}-65a-142$, $192a^{24}-364a^{23}+107a^{22}+322a^{21}-420a^{20}-5a^{19}+496a^{18}-418a^{17}-187a^{16}+649a^{15}-373a^{14}-430a^{13}+814a^{12}-211a^{11}-733a^{10}+902a^{9}+25a^{8}-1095a^{7}+936a^{6}+444a^{5}-1459a^{4}+802a^{3}+958a^{2}-1827a+874$, $32a^{24}+10a^{23}-9a^{22}-32a^{21}-46a^{20}-14a^{19}-2a^{18}+44a^{17}+39a^{16}+46a^{15}-a^{14}-43a^{13}-51a^{12}-64a^{11}+a^{10}+12a^{9}+93a^{8}+60a^{7}+35a^{6}-27a^{5}-88a^{4}-71a^{3}-99a^{2}+56a+122$, $152a^{24}+204a^{23}+247a^{22}+258a^{21}+271a^{20}+286a^{19}+251a^{18}+197a^{17}+126a^{16}+70a^{15}+5a^{14}-117a^{13}-206a^{12}-280a^{11}-347a^{10}-423a^{9}-514a^{8}-491a^{7}-452a^{6}-423a^{5}-355a^{4}-258a^{3}-51a^{2}+73a+466$
| 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: | \( 12233819879819124 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 12233819879819124 \cdot 1}{2\cdot\sqrt{5293955920384129865607305662082268504564193838180857}}\cr\approx \mathstrut & 0.636546803906511 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 + 2*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 + 2*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 + 2*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 + 2*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 $S_{25}$ 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 | $20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.8.0.1}{8} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}$ | $17{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $23{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | $25$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(137\)
| $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.2.1.2 | $x^{2} + 411$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 137.7.0.1 | $x^{7} + x + 134$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 137.13.0.1 | $x^{13} + 14 x + 134$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
|
\(3187\)
| $\Q_{3187}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{3187}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{3187}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
|
\(319234466347\)
| 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 $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(379\!\cdots\!049\)
| $\Q_{37\!\cdots\!49}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ |