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Properties

Label 25.5.298...000.5
Degree $25$
Signature $(5, 10)$
Discriminant $2.980\times 10^{37}$
Root discriminant \(31.55\)
Ramified primes $2,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_5^2:F_5$ (as 25T35)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^25 + 15*x^23 + 140*x^21 - 35*x^20 + 350*x^19 - 570*x^18 - 875*x^17 - 1320*x^16 - 4975*x^15 + 5200*x^14 - 12040*x^13 + 25025*x^12 - 25340*x^11 + 36260*x^10 - 30000*x^9 + 13870*x^8 - 26600*x^7 - 880*x^6 - 18195*x^5 + 22150*x^4 + 4365*x^3 - 5900*x^2 + 3140*x + 1273)
 
Copy content gp:K = bnfinit(y^25 + 15*y^23 + 140*y^21 - 35*y^20 + 350*y^19 - 570*y^18 - 875*y^17 - 1320*y^16 - 4975*y^15 + 5200*y^14 - 12040*y^13 + 25025*y^12 - 25340*y^11 + 36260*y^10 - 30000*y^9 + 13870*y^8 - 26600*y^7 - 880*y^6 - 18195*y^5 + 22150*y^4 + 4365*y^3 - 5900*y^2 + 3140*y + 1273, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 + 15*x^23 + 140*x^21 - 35*x^20 + 350*x^19 - 570*x^18 - 875*x^17 - 1320*x^16 - 4975*x^15 + 5200*x^14 - 12040*x^13 + 25025*x^12 - 25340*x^11 + 36260*x^10 - 30000*x^9 + 13870*x^8 - 26600*x^7 - 880*x^6 - 18195*x^5 + 22150*x^4 + 4365*x^3 - 5900*x^2 + 3140*x + 1273);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^25 + 15*x^23 + 140*x^21 - 35*x^20 + 350*x^19 - 570*x^18 - 875*x^17 - 1320*x^16 - 4975*x^15 + 5200*x^14 - 12040*x^13 + 25025*x^12 - 25340*x^11 + 36260*x^10 - 30000*x^9 + 13870*x^8 - 26600*x^7 - 880*x^6 - 18195*x^5 + 22150*x^4 + 4365*x^3 - 5900*x^2 + 3140*x + 1273)
 

\( x^{25} + 15 x^{23} + 140 x^{21} - 35 x^{20} + 350 x^{19} - 570 x^{18} - 875 x^{17} - 1320 x^{16} + \cdots + 1273 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $25$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(5, 10)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(29802322387695312500000000000000000000\) \(\medspace = 2^{20}\cdot 5^{45}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(31.55\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(5\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_5$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{19}a^{23}-\frac{9}{19}a^{22}+\frac{8}{19}a^{21}-\frac{2}{19}a^{20}+\frac{5}{19}a^{19}+\frac{1}{19}a^{18}-\frac{4}{19}a^{17}+\frac{5}{19}a^{16}+\frac{2}{19}a^{15}+\frac{8}{19}a^{14}+\frac{2}{19}a^{13}-\frac{6}{19}a^{12}-\frac{2}{19}a^{11}-\frac{3}{19}a^{10}+\frac{6}{19}a^{8}+\frac{4}{19}a^{7}+\frac{6}{19}a^{6}-\frac{7}{19}a^{5}+\frac{4}{19}a^{4}-\frac{2}{19}a^{3}+\frac{4}{19}a^{2}+\frac{2}{19}a$, $\frac{1}{59\cdots 37}a^{24}+\frac{72\cdots 96}{59\cdots 37}a^{23}+\frac{26\cdots 13}{59\cdots 37}a^{22}+\frac{39\cdots 43}{59\cdots 37}a^{21}+\frac{93\cdots 22}{59\cdots 37}a^{20}-\frac{62\cdots 30}{59\cdots 37}a^{19}+\frac{24\cdots 78}{59\cdots 37}a^{18}-\frac{23\cdots 03}{59\cdots 37}a^{17}-\frac{11\cdots 11}{59\cdots 37}a^{16}+\frac{12\cdots 72}{59\cdots 37}a^{15}-\frac{29\cdots 54}{59\cdots 37}a^{14}-\frac{13\cdots 30}{59\cdots 37}a^{13}-\frac{23\cdots 49}{59\cdots 37}a^{12}-\frac{82\cdots 53}{59\cdots 37}a^{11}-\frac{19\cdots 85}{59\cdots 37}a^{10}+\frac{19\cdots 99}{59\cdots 37}a^{9}+\frac{27\cdots 38}{59\cdots 37}a^{8}+\frac{10\cdots 58}{59\cdots 37}a^{7}+\frac{15\cdots 08}{59\cdots 37}a^{6}+\frac{26\cdots 89}{59\cdots 37}a^{5}-\frac{16\cdots 19}{59\cdots 37}a^{4}-\frac{25\cdots 46}{59\cdots 37}a^{3}+\frac{25\cdots 85}{59\cdots 37}a^{2}-\frac{11\cdots 58}{59\cdots 37}a+\frac{10\cdots 41}{31\cdots 23}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $14$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{21\cdots 42}{59\cdots 37}a^{24}+\frac{42\cdots 21}{31\cdots 23}a^{23}+\frac{32\cdots 78}{59\cdots 37}a^{22}+\frac{12\cdots 17}{59\cdots 37}a^{21}+\frac{30\cdots 68}{59\cdots 37}a^{20}+\frac{10\cdots 44}{59\cdots 37}a^{19}+\frac{53\cdots 76}{59\cdots 37}a^{18}+\frac{20\cdots 53}{59\cdots 37}a^{17}-\frac{63\cdots 15}{59\cdots 37}a^{16}-\frac{88\cdots 23}{59\cdots 37}a^{15}-\frac{23\cdots 60}{59\cdots 37}a^{14}-\frac{30\cdots 56}{59\cdots 37}a^{13}+\frac{10\cdots 15}{59\cdots 37}a^{12}-\frac{56\cdots 95}{59\cdots 37}a^{11}+\frac{15\cdots 68}{59\cdots 37}a^{10}-\frac{16\cdots 97}{59\cdots 37}a^{9}+\frac{27\cdots 92}{59\cdots 37}a^{8}-\frac{30\cdots 82}{59\cdots 37}a^{7}+\frac{16\cdots 11}{59\cdots 37}a^{6}-\frac{27\cdots 77}{59\cdots 37}a^{5}+\frac{18\cdots 56}{59\cdots 37}a^{4}-\frac{84\cdots 30}{59\cdots 37}a^{3}+\frac{58\cdots 02}{31\cdots 23}a^{2}+\frac{13\cdots 42}{59\cdots 37}a-\frac{29\cdots 73}{31\cdots 23}$, $\frac{19\cdots 25}{59\cdots 37}a^{24}+\frac{92\cdots 16}{59\cdots 37}a^{23}+\frac{28\cdots 40}{59\cdots 37}a^{22}+\frac{14\cdots 66}{59\cdots 37}a^{21}+\frac{26\cdots 67}{59\cdots 37}a^{20}+\frac{69\cdots 54}{59\cdots 37}a^{19}+\frac{64\cdots 24}{59\cdots 37}a^{18}-\frac{71\cdots 89}{59\cdots 37}a^{17}-\frac{23\cdots 62}{59\cdots 37}a^{16}-\frac{33\cdots 60}{59\cdots 37}a^{15}-\frac{11\cdots 13}{59\cdots 37}a^{14}+\frac{57\cdots 43}{59\cdots 37}a^{13}-\frac{18\cdots 23}{59\cdots 37}a^{12}+\frac{37\cdots 50}{59\cdots 37}a^{11}-\frac{21\cdots 79}{59\cdots 37}a^{10}+\frac{35\cdots 52}{59\cdots 37}a^{9}-\frac{73\cdots 43}{59\cdots 37}a^{8}-\frac{33\cdots 52}{59\cdots 37}a^{7}+\frac{62\cdots 93}{59\cdots 37}a^{6}-\frac{80\cdots 10}{59\cdots 37}a^{5}+\frac{11\cdots 69}{31\cdots 23}a^{4}-\frac{57\cdots 01}{59\cdots 37}a^{3}+\frac{79\cdots 21}{59\cdots 37}a^{2}-\frac{16\cdots 72}{59\cdots 37}a+\frac{28\cdots 42}{31\cdots 23}$, $\frac{12\cdots 83}{59\cdots 37}a^{24}-\frac{22\cdots 90}{59\cdots 37}a^{23}-\frac{18\cdots 33}{59\cdots 37}a^{22}+\frac{84\cdots 03}{59\cdots 37}a^{21}-\frac{91\cdots 23}{31\cdots 23}a^{20}+\frac{46\cdots 62}{59\cdots 37}a^{19}-\frac{44\cdots 46}{59\cdots 37}a^{18}+\frac{75\cdots 50}{59\cdots 37}a^{17}+\frac{10\cdots 58}{59\cdots 37}a^{16}+\frac{18\cdots 93}{59\cdots 37}a^{15}+\frac{60\cdots 26}{59\cdots 37}a^{14}-\frac{66\cdots 53}{59\cdots 37}a^{13}+\frac{14\cdots 36}{59\cdots 37}a^{12}-\frac{33\cdots 95}{59\cdots 37}a^{11}+\frac{33\cdots 74}{59\cdots 37}a^{10}-\frac{49\cdots 55}{59\cdots 37}a^{9}+\frac{24\cdots 90}{31\cdots 23}a^{8}-\frac{25\cdots 02}{59\cdots 37}a^{7}+\frac{45\cdots 04}{59\cdots 37}a^{6}-\frac{83\cdots 22}{59\cdots 37}a^{5}+\frac{27\cdots 84}{59\cdots 37}a^{4}-\frac{36\cdots 28}{59\cdots 37}a^{3}-\frac{75\cdots 46}{59\cdots 37}a^{2}+\frac{16\cdots 06}{59\cdots 37}a-\frac{39\cdots 21}{31\cdots 23}$, $\frac{11\cdots 93}{59\cdots 37}a^{24}-\frac{97\cdots 65}{59\cdots 37}a^{23}+\frac{16\cdots 02}{59\cdots 37}a^{22}-\frac{15\cdots 16}{59\cdots 37}a^{21}+\frac{15\cdots 35}{59\cdots 37}a^{20}-\frac{18\cdots 66}{59\cdots 37}a^{19}+\frac{39\cdots 35}{59\cdots 37}a^{18}-\frac{10\cdots 62}{59\cdots 37}a^{17}-\frac{52\cdots 19}{59\cdots 37}a^{16}-\frac{68\cdots 71}{59\cdots 37}a^{15}-\frac{39\cdots 04}{59\cdots 37}a^{14}+\frac{11\cdots 63}{59\cdots 37}a^{13}-\frac{16\cdots 71}{59\cdots 37}a^{12}+\frac{41\cdots 54}{59\cdots 37}a^{11}-\frac{51\cdots 70}{59\cdots 37}a^{10}+\frac{66\cdots 66}{59\cdots 37}a^{9}-\frac{72\cdots 96}{59\cdots 37}a^{8}+\frac{45\cdots 46}{59\cdots 37}a^{7}-\frac{47\cdots 81}{59\cdots 37}a^{6}+\frac{26\cdots 92}{59\cdots 37}a^{5}-\frac{10\cdots 55}{59\cdots 37}a^{4}+\frac{45\cdots 35}{59\cdots 37}a^{3}+\frac{14\cdots 19}{59\cdots 37}a^{2}-\frac{81\cdots 02}{59\cdots 37}a+\frac{42\cdots 69}{31\cdots 23}$, $\frac{32\cdots 10}{59\cdots 37}a^{24}+\frac{22\cdots 70}{59\cdots 37}a^{23}-\frac{47\cdots 03}{59\cdots 37}a^{22}+\frac{17\cdots 34}{31\cdots 23}a^{21}-\frac{44\cdots 84}{59\cdots 37}a^{20}+\frac{32\cdots 15}{59\cdots 37}a^{19}-\frac{17\cdots 37}{59\cdots 37}a^{18}+\frac{10\cdots 82}{59\cdots 37}a^{17}-\frac{98\cdots 06}{59\cdots 37}a^{16}-\frac{15\cdots 85}{59\cdots 37}a^{15}-\frac{17\cdots 88}{59\cdots 37}a^{14}-\frac{13\cdots 88}{59\cdots 37}a^{13}+\frac{14\cdots 69}{59\cdots 37}a^{12}-\frac{35\cdots 65}{59\cdots 37}a^{11}+\frac{65\cdots 42}{59\cdots 37}a^{10}-\frac{68\cdots 14}{59\cdots 37}a^{9}+\frac{96\cdots 40}{59\cdots 37}a^{8}-\frac{79\cdots 97}{59\cdots 37}a^{7}+\frac{48\cdots 37}{59\cdots 37}a^{6}-\frac{74\cdots 88}{59\cdots 37}a^{5}+\frac{13\cdots 71}{59\cdots 37}a^{4}-\frac{60\cdots 40}{59\cdots 37}a^{3}+\frac{57\cdots 31}{59\cdots 37}a^{2}+\frac{74\cdots 64}{59\cdots 37}a+\frac{17\cdots 58}{31\cdots 23}$, $\frac{16\cdots 62}{59\cdots 37}a^{24}-\frac{34\cdots 34}{59\cdots 37}a^{23}-\frac{23\cdots 53}{59\cdots 37}a^{22}-\frac{53\cdots 71}{59\cdots 37}a^{21}-\frac{21\cdots 57}{59\cdots 37}a^{20}-\frac{45\cdots 06}{59\cdots 37}a^{19}-\frac{34\cdots 99}{59\cdots 37}a^{18}-\frac{54\cdots 81}{59\cdots 37}a^{17}+\frac{36\cdots 77}{59\cdots 37}a^{16}+\frac{40\cdots 64}{59\cdots 37}a^{15}+\frac{12\cdots 99}{59\cdots 37}a^{14}+\frac{93\cdots 78}{59\cdots 37}a^{13}+\frac{68\cdots 03}{59\cdots 37}a^{12}+\frac{14\cdots 51}{59\cdots 37}a^{11}-\frac{59\cdots 58}{59\cdots 37}a^{10}+\frac{68\cdots 36}{59\cdots 37}a^{9}-\frac{12\cdots 73}{59\cdots 37}a^{8}+\frac{13\cdots 83}{59\cdots 37}a^{7}-\frac{67\cdots 36}{59\cdots 37}a^{6}+\frac{12\cdots 63}{59\cdots 37}a^{5}+\frac{87\cdots 58}{59\cdots 37}a^{4}+\frac{44\cdots 96}{59\cdots 37}a^{3}-\frac{10\cdots 97}{59\cdots 37}a^{2}+\frac{38\cdots 88}{59\cdots 37}a-\frac{88\cdots 15}{31\cdots 23}$, $\frac{15\cdots 84}{59\cdots 37}a^{24}-\frac{14\cdots 67}{59\cdots 37}a^{23}+\frac{23\cdots 00}{59\cdots 37}a^{22}-\frac{21\cdots 81}{59\cdots 37}a^{21}+\frac{21\cdots 94}{59\cdots 37}a^{20}-\frac{25\cdots 09}{59\cdots 37}a^{19}+\frac{65\cdots 09}{59\cdots 37}a^{18}-\frac{13\cdots 95}{59\cdots 37}a^{17}-\frac{24\cdots 50}{59\cdots 37}a^{16}-\frac{10\cdots 49}{59\cdots 37}a^{15}-\frac{59\cdots 06}{59\cdots 37}a^{14}+\frac{14\cdots 50}{59\cdots 37}a^{13}-\frac{28\cdots 80}{59\cdots 37}a^{12}+\frac{57\cdots 78}{59\cdots 37}a^{11}-\frac{82\cdots 43}{59\cdots 37}a^{10}+\frac{10\cdots 36}{59\cdots 37}a^{9}-\frac{11\cdots 59}{59\cdots 37}a^{8}+\frac{90\cdots 78}{59\cdots 37}a^{7}-\frac{42\cdots 07}{31\cdots 23}a^{6}+\frac{55\cdots 18}{59\cdots 37}a^{5}-\frac{42\cdots 06}{59\cdots 37}a^{4}+\frac{67\cdots 46}{59\cdots 37}a^{3}-\frac{41\cdots 84}{59\cdots 37}a^{2}-\frac{66\cdots 07}{59\cdots 37}a+\frac{35\cdots 56}{31\cdots 23}$, $\frac{12\cdots 39}{59\cdots 37}a^{24}-\frac{10\cdots 73}{59\cdots 37}a^{23}+\frac{18\cdots 47}{59\cdots 37}a^{22}-\frac{16\cdots 18}{59\cdots 37}a^{21}+\frac{16\cdots 25}{59\cdots 37}a^{20}-\frac{20\cdots 10}{59\cdots 37}a^{19}+\frac{42\cdots 92}{59\cdots 37}a^{18}-\frac{59\cdots 74}{31\cdots 23}a^{17}-\frac{59\cdots 74}{59\cdots 37}a^{16}-\frac{64\cdots 11}{59\cdots 37}a^{15}-\frac{43\cdots 85}{59\cdots 37}a^{14}+\frac{12\cdots 24}{59\cdots 37}a^{13}-\frac{96\cdots 95}{31\cdots 23}a^{12}+\frac{44\cdots 79}{59\cdots 37}a^{11}-\frac{54\cdots 24}{59\cdots 37}a^{10}+\frac{69\cdots 61}{59\cdots 37}a^{9}-\frac{75\cdots 97}{59\cdots 37}a^{8}+\frac{47\cdots 29}{59\cdots 37}a^{7}-\frac{55\cdots 07}{59\cdots 37}a^{6}+\frac{19\cdots 27}{31\cdots 23}a^{5}-\frac{27\cdots 72}{59\cdots 37}a^{4}+\frac{57\cdots 22}{59\cdots 37}a^{3}-\frac{14\cdots 10}{59\cdots 37}a^{2}-\frac{47\cdots 23}{59\cdots 37}a+\frac{34\cdots 45}{31\cdots 23}$, $\frac{12\cdots 69}{59\cdots 37}a^{24}+\frac{14\cdots 16}{59\cdots 37}a^{23}-\frac{19\cdots 51}{59\cdots 37}a^{22}+\frac{23\cdots 22}{59\cdots 37}a^{21}-\frac{17\cdots 45}{59\cdots 37}a^{20}+\frac{66\cdots 41}{59\cdots 37}a^{19}-\frac{43\cdots 19}{59\cdots 37}a^{18}+\frac{77\cdots 29}{59\cdots 37}a^{17}+\frac{10\cdots 88}{59\cdots 37}a^{16}+\frac{13\cdots 39}{59\cdots 37}a^{15}+\frac{31\cdots 48}{31\cdots 23}a^{14}-\frac{79\cdots 77}{59\cdots 37}a^{13}+\frac{15\cdots 97}{59\cdots 37}a^{12}-\frac{32\cdots 83}{59\cdots 37}a^{11}+\frac{35\cdots 91}{59\cdots 37}a^{10}-\frac{43\cdots 45}{59\cdots 37}a^{9}+\frac{37\cdots 37}{59\cdots 37}a^{8}-\frac{82\cdots 31}{59\cdots 37}a^{7}+\frac{16\cdots 80}{59\cdots 37}a^{6}+\frac{13\cdots 87}{59\cdots 37}a^{5}+\frac{17\cdots 69}{59\cdots 37}a^{4}-\frac{24\cdots 03}{59\cdots 37}a^{3}-\frac{30\cdots 28}{59\cdots 37}a^{2}+\frac{18\cdots 39}{59\cdots 37}a-\frac{10\cdots 88}{31\cdots 23}$, $\frac{61\cdots 31}{59\cdots 37}a^{24}+\frac{16\cdots 75}{59\cdots 37}a^{23}+\frac{94\cdots 29}{59\cdots 37}a^{22}+\frac{30\cdots 82}{59\cdots 37}a^{21}+\frac{89\cdots 44}{59\cdots 37}a^{20}-\frac{21\cdots 75}{59\cdots 37}a^{19}+\frac{24\cdots 24}{59\cdots 37}a^{18}-\frac{35\cdots 71}{59\cdots 37}a^{17}-\frac{47\cdots 18}{59\cdots 37}a^{16}-\frac{91\cdots 62}{59\cdots 37}a^{15}-\frac{32\cdots 45}{59\cdots 37}a^{14}+\frac{29\cdots 28}{59\cdots 37}a^{13}-\frac{82\cdots 53}{59\cdots 37}a^{12}+\frac{16\cdots 53}{59\cdots 37}a^{11}-\frac{17\cdots 95}{59\cdots 37}a^{10}+\frac{26\cdots 98}{59\cdots 37}a^{9}-\frac{23\cdots 65}{59\cdots 37}a^{8}+\frac{15\cdots 48}{59\cdots 37}a^{7}-\frac{11\cdots 96}{31\cdots 23}a^{6}+\frac{29\cdots 00}{59\cdots 37}a^{5}-\frac{16\cdots 01}{59\cdots 37}a^{4}+\frac{14\cdots 25}{59\cdots 37}a^{3}-\frac{10\cdots 94}{59\cdots 37}a^{2}+\frac{43\cdots 62}{59\cdots 37}a+\frac{96\cdots 36}{31\cdots 23}$, $\frac{10\cdots 70}{59\cdots 37}a^{24}+\frac{46\cdots 87}{59\cdots 37}a^{23}-\frac{15\cdots 84}{59\cdots 37}a^{22}+\frac{69\cdots 39}{59\cdots 37}a^{21}-\frac{14\cdots 75}{59\cdots 37}a^{20}+\frac{10\cdots 06}{59\cdots 37}a^{19}-\frac{38\cdots 61}{59\cdots 37}a^{18}+\frac{75\cdots 80}{59\cdots 37}a^{17}+\frac{60\cdots 77}{59\cdots 37}a^{16}+\frac{10\cdots 90}{59\cdots 37}a^{15}+\frac{45\cdots 88}{59\cdots 37}a^{14}-\frac{75\cdots 77}{59\cdots 37}a^{13}+\frac{15\cdots 35}{59\cdots 37}a^{12}-\frac{31\cdots 25}{59\cdots 37}a^{11}+\frac{38\cdots 75}{59\cdots 37}a^{10}-\frac{52\cdots 26}{59\cdots 37}a^{9}+\frac{50\cdots 61}{59\cdots 37}a^{8}-\frac{32\cdots 09}{59\cdots 37}a^{7}+\frac{38\cdots 19}{59\cdots 37}a^{6}-\frac{10\cdots 29}{59\cdots 37}a^{5}+\frac{10\cdots 25}{31\cdots 23}a^{4}-\frac{29\cdots 79}{59\cdots 37}a^{3}+\frac{18\cdots 87}{59\cdots 37}a^{2}+\frac{16\cdots 24}{59\cdots 37}a-\frac{62\cdots 92}{31\cdots 23}$, $\frac{40\cdots 18}{59\cdots 37}a^{24}-\frac{60\cdots 69}{59\cdots 37}a^{23}+\frac{66\cdots 98}{59\cdots 37}a^{22}-\frac{87\cdots 33}{59\cdots 37}a^{21}+\frac{64\cdots 61}{59\cdots 37}a^{20}-\frac{95\cdots 08}{59\cdots 37}a^{19}+\frac{23\cdots 17}{59\cdots 37}a^{18}-\frac{42\cdots 42}{59\cdots 37}a^{17}+\frac{13\cdots 44}{59\cdots 37}a^{16}-\frac{24\cdots 68}{59\cdots 37}a^{15}-\frac{18\cdots 50}{59\cdots 37}a^{14}+\frac{42\cdots 84}{59\cdots 37}a^{13}-\frac{10\cdots 11}{59\cdots 37}a^{12}+\frac{19\cdots 86}{59\cdots 37}a^{11}-\frac{29\cdots 16}{59\cdots 37}a^{10}+\frac{39\cdots 25}{59\cdots 37}a^{9}-\frac{41\cdots 13}{59\cdots 37}a^{8}+\frac{34\cdots 06}{59\cdots 37}a^{7}-\frac{28\cdots 12}{59\cdots 37}a^{6}+\frac{18\cdots 78}{59\cdots 37}a^{5}-\frac{22\cdots 86}{59\cdots 37}a^{4}+\frac{19\cdots 27}{59\cdots 37}a^{3}-\frac{15\cdots 70}{59\cdots 37}a^{2}+\frac{33\cdots 18}{59\cdots 37}a+\frac{36\cdots 60}{31\cdots 23}$, $\frac{10\cdots 92}{59\cdots 37}a^{24}+\frac{18\cdots 70}{59\cdots 37}a^{23}-\frac{16\cdots 61}{59\cdots 37}a^{22}+\frac{29\cdots 30}{59\cdots 37}a^{21}-\frac{15\cdots 01}{59\cdots 37}a^{20}+\frac{65\cdots 20}{59\cdots 37}a^{19}-\frac{37\cdots 55}{59\cdots 37}a^{18}+\frac{68\cdots 10}{59\cdots 37}a^{17}+\frac{83\cdots 81}{59\cdots 37}a^{16}+\frac{11\cdots 49}{59\cdots 37}a^{15}+\frac{48\cdots 81}{59\cdots 37}a^{14}-\frac{67\cdots 40}{59\cdots 37}a^{13}+\frac{13\cdots 40}{59\cdots 37}a^{12}-\frac{28\cdots 99}{59\cdots 37}a^{11}+\frac{33\cdots 52}{59\cdots 37}a^{10}-\frac{41\cdots 24}{59\cdots 37}a^{9}+\frac{40\cdots 11}{59\cdots 37}a^{8}-\frac{20\cdots 91}{59\cdots 37}a^{7}+\frac{31\cdots 82}{59\cdots 37}a^{6}-\frac{78\cdots 47}{59\cdots 37}a^{5}+\frac{17\cdots 94}{59\cdots 37}a^{4}-\frac{27\cdots 13}{59\cdots 37}a^{3}+\frac{19\cdots 43}{59\cdots 37}a^{2}+\frac{17\cdots 51}{59\cdots 37}a+\frac{90\cdots 71}{31\cdots 23}$, $\frac{18\cdots 70}{59\cdots 37}a^{24}-\frac{13\cdots 91}{59\cdots 37}a^{23}-\frac{14\cdots 29}{31\cdots 23}a^{22}+\frac{51\cdots 04}{59\cdots 37}a^{21}-\frac{24\cdots 18}{59\cdots 37}a^{20}+\frac{64\cdots 60}{59\cdots 37}a^{19}-\frac{58\cdots 90}{59\cdots 37}a^{18}+\frac{10\cdots 26}{59\cdots 37}a^{17}+\frac{17\cdots 96}{59\cdots 37}a^{16}+\frac{22\cdots 05}{59\cdots 37}a^{15}+\frac{86\cdots 99}{59\cdots 37}a^{14}-\frac{53\cdots 32}{31\cdots 23}a^{13}+\frac{19\cdots 99}{59\cdots 37}a^{12}-\frac{44\cdots 51}{59\cdots 37}a^{11}+\frac{42\cdots 05}{59\cdots 37}a^{10}-\frac{58\cdots 81}{59\cdots 37}a^{9}+\frac{50\cdots 75}{59\cdots 37}a^{8}-\frac{12\cdots 97}{59\cdots 37}a^{7}+\frac{41\cdots 34}{59\cdots 37}a^{6}+\frac{75\cdots 43}{59\cdots 37}a^{5}+\frac{22\cdots 05}{59\cdots 37}a^{4}-\frac{40\cdots 65}{59\cdots 37}a^{3}-\frac{15\cdots 87}{59\cdots 37}a^{2}+\frac{18\cdots 48}{59\cdots 37}a-\frac{18\cdots 09}{31\cdots 23}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 1025726545.4439231 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 5 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{10}\cdot 1025726545.4439231 \cdot 1}{2\cdot\sqrt{29802322387695312500000000000000000000}}\cr\approx \mathstrut & 0.288287098450481 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^25 + 15*x^23 + 140*x^21 - 35*x^20 + 350*x^19 - 570*x^18 - 875*x^17 - 1320*x^16 - 4975*x^15 + 5200*x^14 - 12040*x^13 + 25025*x^12 - 25340*x^11 + 36260*x^10 - 30000*x^9 + 13870*x^8 - 26600*x^7 - 880*x^6 - 18195*x^5 + 22150*x^4 + 4365*x^3 - 5900*x^2 + 3140*x + 1273) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^25 + 15*x^23 + 140*x^21 - 35*x^20 + 350*x^19 - 570*x^18 - 875*x^17 - 1320*x^16 - 4975*x^15 + 5200*x^14 - 12040*x^13 + 25025*x^12 - 25340*x^11 + 36260*x^10 - 30000*x^9 + 13870*x^8 - 26600*x^7 - 880*x^6 - 18195*x^5 + 22150*x^4 + 4365*x^3 - 5900*x^2 + 3140*x + 1273, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 + 15*x^23 + 140*x^21 - 35*x^20 + 350*x^19 - 570*x^18 - 875*x^17 - 1320*x^16 - 4975*x^15 + 5200*x^14 - 12040*x^13 + 25025*x^12 - 25340*x^11 + 36260*x^10 - 30000*x^9 + 13870*x^8 - 26600*x^7 - 880*x^6 - 18195*x^5 + 22150*x^4 + 4365*x^3 - 5900*x^2 + 3140*x + 1273); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^25 + 15*x^23 + 140*x^21 - 35*x^20 + 350*x^19 - 570*x^18 - 875*x^17 - 1320*x^16 - 4975*x^15 + 5200*x^14 - 12040*x^13 + 25025*x^12 - 25340*x^11 + 36260*x^10 - 30000*x^9 + 13870*x^8 - 26600*x^7 - 880*x^6 - 18195*x^5 + 22150*x^4 + 4365*x^3 - 5900*x^2 + 3140*x + 1273); 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

$C_5^2:F_5$ (as 25T35):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 500
The 26 conjugacy class representatives for $C_5^2:F_5$
Character table for $C_5^2:F_5$

Intermediate fields

5.1.50000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 25 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $20{,}\,{\href{/padicField/3.5.0.1}{5} }$ R $20{,}\,{\href{/padicField/7.5.0.1}{5} }$ ${\href{/padicField/11.5.0.1}{5} }^{5}$ $20{,}\,{\href{/padicField/13.5.0.1}{5} }$ $20{,}\,{\href{/padicField/17.5.0.1}{5} }$ ${\href{/padicField/19.2.0.1}{2} }^{10}{,}\,{\href{/padicField/19.1.0.1}{1} }^{5}$ $20{,}\,{\href{/padicField/23.5.0.1}{5} }$ ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.5.0.1}{5} }$ ${\href{/padicField/31.5.0.1}{5} }^{5}$ ${\href{/padicField/37.4.0.1}{4} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{5}$ ${\href{/padicField/41.5.0.1}{5} }^{5}$ $20{,}\,{\href{/padicField/43.5.0.1}{5} }$ $20{,}\,{\href{/padicField/47.5.0.1}{5} }$ $20{,}\,{\href{/padicField/53.5.0.1}{5} }$ ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.5.0.1}{5} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 Deg $25$$5$$5$$20$
\(5\) Copy content Toggle raw display Deg $25$$25$$1$$45$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)