Properties

Label 25.1.993...000.1
Degree $25$
Signature $[1, 12]$
Discriminant $9.937\times 10^{49}$
Root discriminant \(99.97\)
Ramified primes $2,5,79496847501414068010394661$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{25}$ (as 25T211)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 + 5*x - 2)
 
gp: K = bnfinit(y^25 + 5*y - 2, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 + 5*x - 2);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 + 5*x - 2)
 

\( x^{25} + 5x - 2 \) Copy content Toggle raw display

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:   \(99371059376767585012993326250000000000000000000000\) \(\medspace = 2^{22}\cdot 5^{25}\cdot 79496847501414068010394661\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(99.97\)
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:   \(2\), \(5\), \(79496847501414068010394661\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  $\Q(\sqrt{39748\!\cdots\!73305}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{23}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

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 $2$) Copy content Toggle raw display
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:   $a^{7}+a^{5}+a^{3}+a+1$, $18a^{24}-18a^{23}-2a^{22}+23a^{21}-17a^{20}-12a^{19}+32a^{18}-15a^{17}-25a^{16}+38a^{15}-a^{14}-43a^{13}+35a^{12}+21a^{11}-58a^{10}+28a^{9}+39a^{8}-66a^{7}+12a^{6}+69a^{5}-73a^{4}-24a^{3}+110a^{2}-65a+21$, $84a^{24}+28a^{23}+3a^{22}-8a^{21}-13a^{20}-14a^{19}-13a^{18}-9a^{17}-2a^{16}+7a^{15}+16a^{14}+23a^{13}+25a^{12}+22a^{11}+16a^{10}+7a^{9}-2a^{8}-12a^{7}-22a^{6}-32a^{5}-37a^{4}-37a^{3}-26a^{2}-10a+431$, $14a^{24}+12a^{23}+9a^{22}+6a^{21}+3a^{20}-3a^{18}-5a^{17}-8a^{16}-12a^{15}-18a^{14}-22a^{13}-25a^{12}-25a^{11}-25a^{10}-21a^{9}-17a^{8}-13a^{7}-16a^{6}-21a^{5}-29a^{4}-35a^{3}-43a^{2}-46a+27$, $12a^{24}-20a^{23}-a^{22}+30a^{21}+10a^{20}-29a^{19}-19a^{18}+29a^{17}+30a^{16}-28a^{15}-40a^{14}+22a^{13}+50a^{12}-9a^{11}-64a^{10}-a^{9}+69a^{8}+19a^{7}-73a^{6}-42a^{5}+76a^{4}+68a^{3}-76a^{2}-89a+115$, $9a^{24}-5a^{23}+21a^{22}-16a^{21}+5a^{20}-21a^{19}+20a^{18}-a^{17}+16a^{16}-24a^{15}-3a^{14}-3a^{13}+20a^{12}+11a^{11}-16a^{10}-13a^{9}-19a^{8}+47a^{7}-10a^{6}+26a^{5}-74a^{4}+41a^{3}-38a^{2}+105a-37$, $4a^{24}-a^{22}+4a^{21}-6a^{20}-4a^{19}+a^{18}-7a^{17}-3a^{16}-3a^{14}+5a^{13}+2a^{12}-a^{11}+16a^{10}-a^{9}-a^{8}+12a^{7}-2a^{6}-13a^{5}+9a^{4}-19a^{3}-10a+7$, $8a^{24}-5a^{23}+20a^{22}+8a^{21}+7a^{20}+11a^{19}-5a^{18}-2a^{17}+22a^{16}+a^{15}-4a^{14}+11a^{13}-36a^{12}+15a^{11}+7a^{10}-28a^{9}+5a^{8}-29a^{7}-28a^{6}+11a^{5}-16a^{4}-61a^{3}+16a^{2}-58a+25$, $136a^{24}+50a^{23}+9a^{22}+14a^{21}+7a^{20}+31a^{19}+16a^{18}+11a^{17}-11a^{16}-27a^{15}-12a^{14}-25a^{13}+11a^{12}-8a^{11}+32a^{10}+25a^{9}+29a^{8}+12a^{7}-47a^{6}-26a^{5}-65a^{4}+22a^{3}-a^{2}+49a+715$, $68a^{24}+67a^{23}+73a^{22}+49a^{21}+24a^{20}-3a^{19}-52a^{18}-69a^{17}-102a^{16}-110a^{15}-89a^{14}-82a^{13}-19a^{12}+19a^{11}+70a^{10}+134a^{9}+133a^{8}+171a^{7}+135a^{6}+84a^{5}+51a^{4}-72a^{3}-107a^{2}-191a+95$, $20a^{24}-24a^{23}+15a^{22}+21a^{21}-47a^{20}+22a^{19}+25a^{18}-39a^{17}+26a^{16}-12a^{15}-20a^{14}+66a^{13}-56a^{12}-29a^{11}+92a^{10}-63a^{9}-8a^{8}+62a^{7}-92a^{6}+66a^{5}+42a^{4}-139a^{3}+109a^{2}+9a-7$, $19a^{24}+2a^{23}-2a^{22}-27a^{21}-7a^{20}-8a^{19}+11a^{18}+58a^{17}-19a^{16}-12a^{15}-27a^{14}-32a^{13}+36a^{12}-12a^{11}+84a^{10}-2a^{9}-78a^{8}-a^{7}-56a^{6}+64a^{5}+25a^{4}+37a^{3}+62a^{2}-151a+47$ Copy content Toggle raw display (assuming GRH)
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:  \( 8545612733538318.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 8545612733538318.0 \cdot 1}{2\cdot\sqrt{99371059376767585012993326250000000000000000000000}}\cr\approx \mathstrut & 3.24542567918385 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^25 + 5*x - 2)
 
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 - 2, 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 - 2);
 
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 - 2);
 
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

$S_{25}$ (as 25T211):

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 ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ R ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ $17{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ $20{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ $25$ ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ $19{,}\,{\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ $16{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ $24{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ $18{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.9.0.1}{9} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.7.6.1$x^{7} + 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
Deg $16$$8$$2$$16$
\(5\) Copy content Toggle raw display Deg $25$$25$$1$$25$
\(794\!\cdots\!661\) Copy content Toggle raw display $\Q_{79\!\cdots\!61}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $13$$1$$13$$0$$C_{13}$$[\ ]^{13}$