Properties

Label 25.3.158...375.1
Degree $25$
Signature $[3, 11]$
Discriminant $-1.590\times 10^{49}$
Root discriminant \(92.91\)
Ramified primes $3,5,409,11545399656835619219$
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 - 3)
 
gp: K = bnfinit(y^25 - 5*y - 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 5*x - 3);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 5*x - 3)
 

\( x^{25} - 5x - 3 \) 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:  $[3, 11]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-15898366049400615492894758682139527797698974609375\) \(\medspace = -\,3^{24}\cdot 5^{23}\cdot 409\cdot 11545399656835619219\) 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:  \(92.91\)
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:   \(3\), \(5\), \(409\), \(11545399656835619219\) 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:  $\Q(\sqrt{-23610\!\cdots\!02855}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

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

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

Monogenic:  Not computed
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:  $13$
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:   $a+1$, $a^{23}-a^{22}-a^{21}+a^{20}-a^{18}+a^{16}+a^{13}+a^{12}-a^{11}+a^{10}+3a^{9}-a^{8}-2a^{7}+4a^{6}-6a^{4}+2a^{3}+5a^{2}-7a-5$, $\frac{7}{5}a^{24}-\frac{19}{5}a^{23}+\frac{18}{5}a^{22}-\frac{6}{5}a^{21}-\frac{13}{5}a^{20}+\frac{26}{5}a^{19}-\frac{22}{5}a^{18}-\frac{1}{5}a^{17}+\frac{22}{5}a^{16}-\frac{29}{5}a^{15}+\frac{18}{5}a^{14}+\frac{4}{5}a^{13}-\frac{28}{5}a^{12}+\frac{31}{5}a^{11}-\frac{17}{5}a^{10}-\frac{6}{5}a^{9}+\frac{22}{5}a^{8}-\frac{34}{5}a^{7}+\frac{23}{5}a^{6}+\frac{4}{5}a^{5}-\frac{33}{5}a^{4}+\frac{41}{5}a^{3}-\frac{37}{5}a^{2}-\frac{6}{5}a+\frac{22}{5}$, $\frac{33}{5}a^{24}+\frac{14}{5}a^{23}-\frac{43}{5}a^{22}+\frac{6}{5}a^{21}+\frac{43}{5}a^{20}-\frac{31}{5}a^{19}-\frac{33}{5}a^{18}+\frac{56}{5}a^{17}+\frac{18}{5}a^{16}-\frac{71}{5}a^{15}+\frac{12}{5}a^{14}+\frac{71}{5}a^{13}-\frac{57}{5}a^{12}-\frac{51}{5}a^{11}+\frac{97}{5}a^{10}+\frac{6}{5}a^{9}-\frac{117}{5}a^{8}+\frac{44}{5}a^{7}+\frac{107}{5}a^{6}-\frac{89}{5}a^{5}-\frac{62}{5}a^{4}+\frac{144}{5}a^{3}-\frac{3}{5}a^{2}-\frac{184}{5}a-\frac{82}{5}$, $\frac{63}{5}a^{24}+\frac{69}{5}a^{23}+\frac{62}{5}a^{22}+\frac{26}{5}a^{21}-\frac{17}{5}a^{20}-\frac{56}{5}a^{19}-\frac{98}{5}a^{18}-\frac{124}{5}a^{17}-\frac{102}{5}a^{16}-\frac{51}{5}a^{15}+\frac{17}{5}a^{14}+\frac{96}{5}a^{13}+\frac{163}{5}a^{12}+\frac{194}{5}a^{11}+\frac{177}{5}a^{10}+\frac{96}{5}a^{9}-\frac{17}{5}a^{8}-\frac{141}{5}a^{7}-\frac{268}{5}a^{6}-\frac{334}{5}a^{5}-\frac{287}{5}a^{4}-\frac{176}{5}a^{3}-\frac{3}{5}a^{2}+\frac{231}{5}a+\frac{118}{5}$, $\frac{22}{5}a^{24}-\frac{9}{5}a^{23}+\frac{3}{5}a^{22}-\frac{16}{5}a^{21}+\frac{22}{5}a^{20}-\frac{14}{5}a^{19}-\frac{2}{5}a^{18}-\frac{21}{5}a^{17}+\frac{27}{5}a^{16}-\frac{9}{5}a^{15}+\frac{3}{5}a^{14}-\frac{16}{5}a^{13}+\frac{42}{5}a^{12}-\frac{4}{5}a^{11}-\frac{2}{5}a^{10}-\frac{31}{5}a^{9}+\frac{37}{5}a^{8}-\frac{14}{5}a^{7}-\frac{17}{5}a^{6}-\frac{46}{5}a^{5}+\frac{47}{5}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{31}{5}a-\frac{28}{5}$, $15a^{24}+6a^{23}-22a^{22}+18a^{21}+8a^{20}-28a^{19}+18a^{18}+12a^{17}-33a^{16}+21a^{15}+18a^{14}-39a^{13}+22a^{12}+24a^{11}-51a^{10}+22a^{9}+32a^{8}-56a^{7}+26a^{6}+44a^{5}-69a^{4}+22a^{3}+54a^{2}-83a-56$, $\frac{4}{5}a^{24}-\frac{8}{5}a^{23}-\frac{14}{5}a^{22}+\frac{13}{5}a^{21}+\frac{19}{5}a^{20}-\frac{18}{5}a^{19}-\frac{9}{5}a^{18}+\frac{3}{5}a^{17}+\frac{4}{5}a^{16}+\frac{17}{5}a^{15}+\frac{1}{5}a^{14}-\frac{42}{5}a^{13}+\frac{9}{5}a^{12}+\frac{42}{5}a^{11}-\frac{4}{5}a^{10}-\frac{22}{5}a^{9}-\frac{11}{5}a^{8}-\frac{18}{5}a^{7}+\frac{51}{5}a^{6}+\frac{43}{5}a^{5}-\frac{66}{5}a^{4}-\frac{53}{5}a^{3}+\frac{51}{5}a^{2}+\frac{33}{5}a-\frac{1}{5}$, $\frac{33}{5}a^{24}-\frac{21}{5}a^{23}+\frac{17}{5}a^{22}-\frac{9}{5}a^{21}-\frac{7}{5}a^{20}-\frac{11}{5}a^{19}-\frac{3}{5}a^{18}-\frac{9}{5}a^{17}+\frac{23}{5}a^{16}-\frac{6}{5}a^{15}+\frac{12}{5}a^{14}+\frac{1}{5}a^{13}-\frac{2}{5}a^{12}+\frac{9}{5}a^{11}+\frac{12}{5}a^{10}-\frac{19}{5}a^{9}+\frac{3}{5}a^{8}-\frac{46}{5}a^{7}-\frac{13}{5}a^{6}+\frac{6}{5}a^{5}+\frac{13}{5}a^{4}+\frac{29}{5}a^{3}+\frac{27}{5}a^{2}-\frac{19}{5}a-\frac{127}{5}$, $\frac{18}{5}a^{24}-\frac{6}{5}a^{23}+\frac{2}{5}a^{22}+\frac{1}{5}a^{21}+\frac{18}{5}a^{20}-\frac{16}{5}a^{19}+\frac{17}{5}a^{18}+\frac{11}{5}a^{17}-\frac{17}{5}a^{16}+\frac{19}{5}a^{15}-\frac{3}{5}a^{14}-\frac{9}{5}a^{13}-\frac{12}{5}a^{12}+\frac{19}{5}a^{11}-\frac{53}{5}a^{10}+\frac{11}{5}a^{9}-\frac{12}{5}a^{8}-\frac{41}{5}a^{7}-\frac{8}{5}a^{6}+\frac{1}{5}a^{5}-\frac{42}{5}a^{4}-\frac{16}{5}a^{3}+\frac{42}{5}a^{2}-\frac{69}{5}a-\frac{52}{5}$, $\frac{9}{5}a^{24}-\frac{23}{5}a^{23}-\frac{9}{5}a^{22}-\frac{12}{5}a^{21}-\frac{31}{5}a^{20}-\frac{3}{5}a^{19}-\frac{34}{5}a^{18}-\frac{7}{5}a^{17}-\frac{11}{5}a^{16}-\frac{18}{5}a^{15}+\frac{21}{5}a^{14}-\frac{17}{5}a^{13}+\frac{39}{5}a^{12}+\frac{12}{5}a^{11}+\frac{41}{5}a^{10}+\frac{58}{5}a^{9}+\frac{29}{5}a^{8}+\frac{82}{5}a^{7}+\frac{26}{5}a^{6}+\frac{63}{5}a^{5}+\frac{49}{5}a^{4}-\frac{3}{5}a^{3}+\frac{46}{5}a^{2}-\frac{72}{5}a-\frac{61}{5}$, $\frac{17}{5}a^{24}-\frac{19}{5}a^{23}+\frac{3}{5}a^{22}+\frac{19}{5}a^{21}-\frac{13}{5}a^{20}-\frac{4}{5}a^{19}+\frac{28}{5}a^{18}-\frac{26}{5}a^{17}+\frac{2}{5}a^{16}+\frac{11}{5}a^{15}-\frac{12}{5}a^{14}-\frac{21}{5}a^{13}+\frac{27}{5}a^{12}-\frac{9}{5}a^{11}-\frac{27}{5}a^{10}+\frac{54}{5}a^{9}-\frac{13}{5}a^{8}-\frac{24}{5}a^{7}+\frac{53}{5}a^{6}-\frac{6}{5}a^{5}-\frac{68}{5}a^{4}+\frac{81}{5}a^{3}-\frac{47}{5}a^{2}-\frac{66}{5}a+\frac{2}{5}$, $\frac{78}{5}a^{24}-\frac{6}{5}a^{23}-\frac{73}{5}a^{22}+\frac{101}{5}a^{21}-\frac{77}{5}a^{20}-\frac{1}{5}a^{19}+\frac{107}{5}a^{18}-\frac{149}{5}a^{17}+\frac{83}{5}a^{16}+\frac{29}{5}a^{15}-\frac{118}{5}a^{14}+\frac{166}{5}a^{13}-\frac{112}{5}a^{12}-\frac{51}{5}a^{11}+\frac{177}{5}a^{10}-\frac{184}{5}a^{9}+\frac{98}{5}a^{8}+\frac{64}{5}a^{7}-\frac{248}{5}a^{6}+\frac{266}{5}a^{5}-\frac{72}{5}a^{4}-\frac{136}{5}a^{3}+\frac{262}{5}a^{2}-\frac{294}{5}a-\frac{262}{5}$ 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:  \( 1137438749492350.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{11}\cdot 1137438749492350.0 \cdot 1}{2\cdot\sqrt{15898366049400615492894758682139527797698974609375}}\cr\approx \mathstrut & 0.687527884212649 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x - 3)
 
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 - 3, 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 - 3);
 
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 - 3);
 
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 ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ R R ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ $25$ $20{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ $18{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ $22{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ $15{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ $24{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.12.12.14$x^{12} + 54 x^{10} - 240 x^{9} - 1935 x^{8} + 7236 x^{7} + 45198 x^{6} + 60156 x^{5} + 16686 x^{4} - 13824 x^{3} + 5184 x^{2} - 972 x + 81$$3$$4$$12$12T173$[3/2, 3/2, 3/2, 3/2]_{2}^{4}$
3.12.12.12$x^{12} - 12 x^{11} + 156 x^{10} - 1104 x^{9} + 3420 x^{8} + 972 x^{7} - 1890 x^{6} + 1404 x^{5} + 1944 x^{4} + 432 x^{3} + 324 x^{2} + 324 x + 81$$3$$4$$12$12T41$[3/2, 3/2]_{2}^{4}$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
Deg $24$$24$$1$$23$
\(409\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $7$$1$$7$$0$$C_7$$[\ ]^{7}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
\(11545399656835619219\) Copy content Toggle raw display $\Q_{11545399656835619219}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{11545399656835619219}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $14$$1$$14$$0$$C_{14}$$[\ ]^{14}$