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Properties

Label 25.5.298...000.4
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 - 45*x^22 + 95*x^21 - 235*x^20 + 500*x^19 - 760*x^18 + 605*x^17 - 2330*x^16 - 1835*x^15 - 1375*x^14 - 5310*x^13 + 13755*x^12 - 12555*x^11 + 11755*x^10 - 13250*x^9 + 1465*x^8 + 3355*x^7 - 3655*x^6 + 3280*x^5 - 2125*x^4 + 215*x^3 - 295*x^2 - 55*x + 9)
 
Copy content gp:K = bnfinit(y^25 + 15*y^23 - 45*y^22 + 95*y^21 - 235*y^20 + 500*y^19 - 760*y^18 + 605*y^17 - 2330*y^16 - 1835*y^15 - 1375*y^14 - 5310*y^13 + 13755*y^12 - 12555*y^11 + 11755*y^10 - 13250*y^9 + 1465*y^8 + 3355*y^7 - 3655*y^6 + 3280*y^5 - 2125*y^4 + 215*y^3 - 295*y^2 - 55*y + 9, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 + 15*x^23 - 45*x^22 + 95*x^21 - 235*x^20 + 500*x^19 - 760*x^18 + 605*x^17 - 2330*x^16 - 1835*x^15 - 1375*x^14 - 5310*x^13 + 13755*x^12 - 12555*x^11 + 11755*x^10 - 13250*x^9 + 1465*x^8 + 3355*x^7 - 3655*x^6 + 3280*x^5 - 2125*x^4 + 215*x^3 - 295*x^2 - 55*x + 9);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^25 + 15*x^23 - 45*x^22 + 95*x^21 - 235*x^20 + 500*x^19 - 760*x^18 + 605*x^17 - 2330*x^16 - 1835*x^15 - 1375*x^14 - 5310*x^13 + 13755*x^12 - 12555*x^11 + 11755*x^10 - 13250*x^9 + 1465*x^8 + 3355*x^7 - 3655*x^6 + 3280*x^5 - 2125*x^4 + 215*x^3 - 295*x^2 - 55*x + 9)
 

\( x^{25} + 15 x^{23} - 45 x^{22} + 95 x^{21} - 235 x^{20} + 500 x^{19} - 760 x^{18} + 605 x^{17} + \cdots + 9 \) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{12}-\frac{2}{5}a^{11}-\frac{1}{5}a^{10}-\frac{2}{5}a^{8}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{14}-\frac{1}{5}a^{12}-\frac{2}{5}a^{11}+\frac{2}{5}a^{10}-\frac{2}{5}a^{9}+\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{2}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{15}a^{15}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{7}{15}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{5}a^{5}-\frac{1}{3}a-\frac{2}{5}$, $\frac{1}{15}a^{16}-\frac{1}{15}a^{13}-\frac{7}{15}a^{12}+\frac{4}{15}a^{11}-\frac{4}{15}a^{10}-\frac{1}{3}a^{9}-\frac{1}{5}a^{8}-\frac{1}{15}a^{7}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{3}-\frac{2}{15}a^{2}+\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{15}a^{17}-\frac{1}{15}a^{14}-\frac{1}{15}a^{13}+\frac{1}{15}a^{12}-\frac{1}{15}a^{11}+\frac{4}{15}a^{10}-\frac{1}{5}a^{9}+\frac{2}{15}a^{8}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{4}{15}a^{3}+\frac{1}{5}a^{2}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{15}a^{18}-\frac{1}{15}a^{14}+\frac{1}{15}a^{13}+\frac{4}{15}a^{12}-\frac{2}{5}a^{11}+\frac{4}{15}a^{10}+\frac{7}{15}a^{9}-\frac{1}{3}a^{8}-\frac{1}{5}a^{7}+\frac{2}{15}a^{6}-\frac{1}{5}a^{5}+\frac{4}{15}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}+\frac{4}{15}a-\frac{2}{5}$, $\frac{1}{15}a^{19}+\frac{1}{15}a^{14}+\frac{1}{15}a^{13}-\frac{7}{15}a^{12}+\frac{2}{15}a^{10}-\frac{2}{15}a^{8}-\frac{1}{15}a^{7}+\frac{1}{3}a^{6}+\frac{1}{15}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{15}a^{2}-\frac{1}{3}a-\frac{1}{5}$, $\frac{1}{15}a^{20}+\frac{1}{15}a^{14}-\frac{1}{15}a^{13}+\frac{7}{15}a^{12}+\frac{2}{15}a^{10}-\frac{7}{15}a^{9}+\frac{7}{15}a^{8}-\frac{4}{15}a^{7}+\frac{1}{3}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{4}{15}a^{3}+\frac{7}{15}a^{2}+\frac{1}{3}a$, $\frac{1}{15}a^{21}-\frac{1}{15}a^{14}+\frac{1}{15}a^{13}-\frac{2}{15}a^{12}-\frac{2}{5}a^{11}+\frac{7}{15}a^{10}+\frac{2}{15}a^{9}-\frac{2}{15}a^{8}-\frac{1}{15}a^{7}-\frac{2}{15}a^{6}+\frac{2}{5}a^{5}+\frac{4}{15}a^{4}+\frac{1}{15}a^{3}-\frac{7}{15}a^{2}+\frac{2}{15}a-\frac{1}{5}$, $\frac{1}{45}a^{22}-\frac{1}{45}a^{21}-\frac{1}{45}a^{17}-\frac{1}{15}a^{14}-\frac{2}{45}a^{13}+\frac{2}{15}a^{12}+\frac{16}{45}a^{11}+\frac{16}{45}a^{10}+\frac{16}{45}a^{9}+\frac{1}{5}a^{8}-\frac{13}{45}a^{7}-\frac{8}{45}a^{6}-\frac{2}{45}a^{5}-\frac{2}{5}a^{4}+\frac{1}{15}a^{3}-\frac{2}{5}a^{2}+\frac{8}{45}a-\frac{1}{5}$, $\frac{1}{45}a^{23}-\frac{1}{45}a^{21}-\frac{1}{45}a^{18}-\frac{1}{45}a^{17}+\frac{4}{45}a^{14}+\frac{4}{45}a^{13}-\frac{17}{45}a^{12}-\frac{16}{45}a^{11}-\frac{19}{45}a^{10}+\frac{22}{45}a^{9}-\frac{19}{45}a^{8}-\frac{1}{15}a^{7}-\frac{4}{45}a^{6}-\frac{2}{45}a^{5}-\frac{2}{15}a^{4}-\frac{1}{3}a^{3}-\frac{19}{45}a^{2}+\frac{11}{45}a-\frac{1}{5}$, $\frac{1}{64\cdots 45}a^{24}+\frac{73\cdots 89}{71\cdots 05}a^{23}+\frac{78\cdots 24}{21\cdots 15}a^{22}+\frac{71\cdots 49}{64\cdots 45}a^{21}+\frac{27\cdots 39}{71\cdots 05}a^{20}+\frac{97\cdots 63}{64\cdots 45}a^{19}+\frac{47\cdots 34}{64\cdots 45}a^{18}+\frac{10\cdots 66}{12\cdots 69}a^{17}-\frac{73\cdots 44}{21\cdots 15}a^{16}+\frac{27\cdots 91}{64\cdots 45}a^{15}+\frac{60\cdots 97}{64\cdots 45}a^{14}-\frac{41\cdots 98}{64\cdots 45}a^{13}+\frac{12\cdots 16}{64\cdots 45}a^{12}+\frac{91\cdots 41}{21\cdots 15}a^{11}-\frac{26\cdots 29}{12\cdots 69}a^{10}-\frac{29\cdots 93}{71\cdots 05}a^{9}-\frac{25\cdots 61}{42\cdots 23}a^{8}-\frac{73\cdots 82}{12\cdots 69}a^{7}+\frac{20\cdots 54}{64\cdots 45}a^{6}-\frac{13\cdots 72}{64\cdots 45}a^{5}-\frac{24\cdots 78}{21\cdots 15}a^{4}+\frac{66\cdots 68}{64\cdots 45}a^{3}+\frac{28\cdots 42}{64\cdots 45}a^{2}-\frac{30\cdots 49}{64\cdots 45}a+\frac{12\cdots 90}{14\cdots 41}$ 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:  No
Index:  Not computed
Inessential primes:  $3$

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:   $\frac{21\cdots 48}{49\cdots 65}a^{24}-\frac{13\cdots 92}{49\cdots 65}a^{23}+\frac{32\cdots 01}{49\cdots 65}a^{22}-\frac{11\cdots 49}{49\cdots 65}a^{21}+\frac{88\cdots 66}{16\cdots 55}a^{20}-\frac{63\cdots 61}{49\cdots 65}a^{19}+\frac{46\cdots 21}{16\cdots 55}a^{18}-\frac{22\cdots 68}{49\cdots 65}a^{17}+\frac{25\cdots 21}{54\cdots 85}a^{16}-\frac{11\cdots 62}{98\cdots 13}a^{15}-\frac{33\cdots 79}{16\cdots 55}a^{14}-\frac{12\cdots 66}{49\cdots 65}a^{13}-\frac{10\cdots 48}{49\cdots 65}a^{12}+\frac{36\cdots 14}{49\cdots 65}a^{11}-\frac{45\cdots 88}{49\cdots 65}a^{10}+\frac{89\cdots 87}{10\cdots 57}a^{9}-\frac{39\cdots 44}{49\cdots 65}a^{8}+\frac{58\cdots 87}{16\cdots 55}a^{7}+\frac{83\cdots 01}{49\cdots 65}a^{6}-\frac{15\cdots 91}{49\cdots 65}a^{5}+\frac{74\cdots 08}{32\cdots 71}a^{4}-\frac{69\cdots 11}{49\cdots 65}a^{3}+\frac{76\cdots 13}{16\cdots 55}a^{2}-\frac{92\cdots 31}{98\cdots 13}a+\frac{50\cdots 54}{54\cdots 85}$, $\frac{18\cdots 66}{64\cdots 45}a^{24}+\frac{68\cdots 88}{12\cdots 69}a^{23}-\frac{26\cdots 34}{64\cdots 45}a^{22}+\frac{13\cdots 16}{64\cdots 45}a^{21}-\frac{10\cdots 14}{21\cdots 15}a^{20}+\frac{14\cdots 75}{12\cdots 69}a^{19}-\frac{15\cdots 32}{64\cdots 45}a^{18}+\frac{27\cdots 96}{64\cdots 45}a^{17}-\frac{99\cdots 01}{21\cdots 15}a^{16}+\frac{51\cdots 56}{64\cdots 45}a^{15}-\frac{67\cdots 24}{12\cdots 69}a^{14}-\frac{65\cdots 99}{64\cdots 45}a^{13}+\frac{16\cdots 84}{21\cdots 15}a^{12}-\frac{86\cdots 82}{12\cdots 69}a^{11}+\frac{21\cdots 14}{21\cdots 15}a^{10}-\frac{50\cdots 54}{71\cdots 05}a^{9}+\frac{39\cdots 29}{64\cdots 45}a^{8}-\frac{27\cdots 46}{64\cdots 45}a^{7}-\frac{45\cdots 93}{14\cdots 41}a^{6}+\frac{27\cdots 91}{64\cdots 45}a^{5}-\frac{39\cdots 21}{21\cdots 15}a^{4}+\frac{53\cdots 42}{64\cdots 45}a^{3}-\frac{20\cdots 43}{64\cdots 45}a^{2}-\frac{15\cdots 78}{64\cdots 45}a-\frac{13\cdots 93}{71\cdots 05}$, $\frac{45\cdots 56}{64\cdots 45}a^{24}+\frac{46\cdots 19}{71\cdots 05}a^{23}+\frac{42\cdots 44}{42\cdots 23}a^{22}+\frac{40\cdots 11}{64\cdots 45}a^{21}-\frac{16\cdots 51}{71\cdots 05}a^{20}+\frac{57\cdots 71}{12\cdots 69}a^{19}-\frac{73\cdots 44}{64\cdots 45}a^{18}+\frac{17\cdots 44}{64\cdots 45}a^{17}-\frac{18\cdots 39}{42\cdots 23}a^{16}+\frac{11\cdots 72}{64\cdots 45}a^{15}-\frac{98\cdots 56}{64\cdots 45}a^{14}-\frac{15\cdots 57}{12\cdots 69}a^{13}-\frac{48\cdots 89}{64\cdots 45}a^{12}-\frac{44\cdots 03}{21\cdots 15}a^{11}+\frac{54\cdots 19}{64\cdots 45}a^{10}-\frac{52\cdots 33}{71\cdots 05}a^{9}+\frac{34\cdots 32}{71\cdots 05}a^{8}-\frac{44\cdots 24}{64\cdots 45}a^{7}+\frac{22\cdots 59}{64\cdots 45}a^{6}+\frac{24\cdots 67}{64\cdots 45}a^{5}-\frac{51\cdots 36}{21\cdots 15}a^{4}+\frac{14\cdots 14}{12\cdots 69}a^{3}-\frac{50\cdots 36}{64\cdots 45}a^{2}-\frac{76\cdots 36}{64\cdots 45}a+\frac{16\cdots 64}{71\cdots 05}$, $\frac{12\cdots 16}{64\cdots 45}a^{24}-\frac{80\cdots 46}{64\cdots 45}a^{23}-\frac{19\cdots 63}{64\cdots 45}a^{22}+\frac{90\cdots 48}{12\cdots 69}a^{21}-\frac{28\cdots 48}{21\cdots 15}a^{20}+\frac{23\cdots 93}{64\cdots 45}a^{19}-\frac{47\cdots 99}{64\cdots 45}a^{18}+\frac{62\cdots 47}{64\cdots 45}a^{17}-\frac{19\cdots 87}{42\cdots 23}a^{16}+\frac{27\cdots 09}{64\cdots 45}a^{15}+\frac{38\cdots 06}{64\cdots 45}a^{14}+\frac{37\cdots 19}{64\cdots 45}a^{13}+\frac{49\cdots 04}{42\cdots 23}a^{12}-\frac{26\cdots 19}{12\cdots 69}a^{11}+\frac{18\cdots 63}{21\cdots 15}a^{10}-\frac{26\cdots 89}{21\cdots 15}a^{9}+\frac{13\cdots 28}{64\cdots 45}a^{8}+\frac{33\cdots 81}{64\cdots 45}a^{7}-\frac{90\cdots 27}{42\cdots 23}a^{6}-\frac{14\cdots 49}{64\cdots 45}a^{5}-\frac{61\cdots 92}{21\cdots 15}a^{4}+\frac{29\cdots 53}{64\cdots 45}a^{3}-\frac{31\cdots 68}{12\cdots 69}a^{2}+\frac{17\cdots 71}{12\cdots 69}a-\frac{17\cdots 64}{14\cdots 41}$, $\frac{13\cdots 82}{71\cdots 05}a^{24}-\frac{67\cdots 51}{42\cdots 23}a^{23}+\frac{18\cdots 47}{64\cdots 45}a^{22}-\frac{57\cdots 83}{64\cdots 45}a^{21}+\frac{39\cdots 71}{21\cdots 15}a^{20}-\frac{32\cdots 92}{71\cdots 05}a^{19}+\frac{69\cdots 01}{71\cdots 05}a^{18}-\frac{93\cdots 16}{64\cdots 45}a^{17}+\frac{78\cdots 07}{71\cdots 05}a^{16}-\frac{18\cdots 17}{42\cdots 23}a^{15}-\frac{14\cdots 71}{42\cdots 23}a^{14}-\frac{98\cdots 56}{64\cdots 45}a^{13}-\frac{66\cdots 54}{71\cdots 05}a^{12}+\frac{17\cdots 79}{64\cdots 45}a^{11}-\frac{15\cdots 56}{64\cdots 45}a^{10}+\frac{12\cdots 57}{64\cdots 45}a^{9}-\frac{32\cdots 03}{14\cdots 41}a^{8}+\frac{74\cdots 27}{64\cdots 45}a^{7}+\frac{13\cdots 30}{12\cdots 69}a^{6}-\frac{96\cdots 09}{12\cdots 69}a^{5}+\frac{95\cdots 66}{21\cdots 15}a^{4}-\frac{19\cdots 17}{71\cdots 05}a^{3}-\frac{23\cdots 48}{71\cdots 05}a^{2}-\frac{67\cdots 29}{64\cdots 45}a-\frac{38\cdots 23}{71\cdots 05}$, $\frac{29\cdots 80}{47\cdots 49}a^{24}-\frac{41\cdots 88}{21\cdots 05}a^{23}+\frac{21\cdots 36}{23\cdots 45}a^{22}-\frac{66\cdots 19}{21\cdots 05}a^{21}+\frac{15\cdots 99}{23\cdots 45}a^{20}-\frac{22\cdots 45}{14\cdots 47}a^{19}+\frac{73\cdots 42}{21\cdots 05}a^{18}-\frac{11\cdots 06}{21\cdots 05}a^{17}+\frac{10\cdots 49}{23\cdots 45}a^{16}-\frac{34\cdots 24}{23\cdots 45}a^{15}-\frac{34\cdots 45}{43\cdots 41}a^{14}-\frac{34\cdots 48}{21\cdots 05}a^{13}-\frac{61\cdots 36}{21\cdots 05}a^{12}+\frac{20\cdots 86}{21\cdots 05}a^{11}-\frac{20\cdots 68}{21\cdots 05}a^{10}+\frac{32\cdots 20}{43\cdots 41}a^{9}-\frac{17\cdots 78}{21\cdots 05}a^{8}+\frac{11\cdots 23}{71\cdots 35}a^{7}+\frac{81\cdots 81}{21\cdots 05}a^{6}-\frac{14\cdots 45}{43\cdots 41}a^{5}+\frac{26\cdots 80}{14\cdots 47}a^{4}-\frac{72\cdots 51}{71\cdots 35}a^{3}+\frac{18\cdots 24}{21\cdots 05}a^{2}+\frac{12\cdots 36}{21\cdots 05}a-\frac{78\cdots 28}{23\cdots 45}$, $\frac{30\cdots 06}{98\cdots 13}a^{24}-\frac{13\cdots 47}{49\cdots 65}a^{23}-\frac{22\cdots 53}{49\cdots 65}a^{22}+\frac{47\cdots 64}{49\cdots 65}a^{21}-\frac{50\cdots 66}{32\cdots 71}a^{20}+\frac{21\cdots 94}{49\cdots 65}a^{19}-\frac{13\cdots 74}{16\cdots 55}a^{18}+\frac{40\cdots 01}{49\cdots 65}a^{17}+\frac{16\cdots 07}{32\cdots 71}a^{16}+\frac{25\cdots 73}{49\cdots 65}a^{15}+\frac{19\cdots 32}{16\cdots 55}a^{14}+\frac{84\cdots 71}{10\cdots 57}a^{13}+\frac{81\cdots 26}{49\cdots 65}a^{12}-\frac{49\cdots 53}{16\cdots 55}a^{11}-\frac{46\cdots 08}{98\cdots 13}a^{10}+\frac{37\cdots 81}{49\cdots 65}a^{9}+\frac{46\cdots 94}{49\cdots 65}a^{8}+\frac{14\cdots 87}{49\cdots 65}a^{7}-\frac{85\cdots 73}{49\cdots 65}a^{6}-\frac{34\cdots 14}{49\cdots 65}a^{5}+\frac{28\cdots 86}{54\cdots 85}a^{4}-\frac{33\cdots 78}{98\cdots 13}a^{3}+\frac{58\cdots 13}{16\cdots 55}a^{2}+\frac{15\cdots 58}{49\cdots 65}a-\frac{44\cdots 31}{54\cdots 85}$, $\frac{26\cdots 86}{71\cdots 05}a^{24}-\frac{52\cdots 22}{64\cdots 45}a^{23}+\frac{35\cdots 71}{64\cdots 45}a^{22}-\frac{22\cdots 14}{12\cdots 69}a^{21}+\frac{80\cdots 87}{21\cdots 15}a^{20}-\frac{19\cdots 89}{21\cdots 15}a^{19}+\frac{12\cdots 79}{64\cdots 45}a^{18}-\frac{19\cdots 93}{64\cdots 45}a^{17}+\frac{53\cdots 23}{21\cdots 15}a^{16}-\frac{18\cdots 59}{21\cdots 15}a^{15}-\frac{33\cdots 14}{64\cdots 45}a^{14}-\frac{28\cdots 10}{12\cdots 69}a^{13}-\frac{11\cdots 86}{64\cdots 45}a^{12}+\frac{23\cdots 87}{42\cdots 23}a^{11}-\frac{11\cdots 99}{21\cdots 15}a^{10}+\frac{28\cdots 28}{64\cdots 45}a^{9}-\frac{31\cdots 74}{64\cdots 45}a^{8}+\frac{67\cdots 42}{64\cdots 45}a^{7}+\frac{38\cdots 91}{21\cdots 15}a^{6}-\frac{10\cdots 19}{64\cdots 45}a^{5}+\frac{21\cdots 13}{21\cdots 15}a^{4}-\frac{15\cdots 81}{21\cdots 15}a^{3}+\frac{77\cdots 16}{64\cdots 45}a^{2}-\frac{28\cdots 22}{64\cdots 45}a+\frac{68\cdots 08}{71\cdots 05}$, $\frac{27\cdots 19}{12\cdots 69}a^{24}-\frac{57\cdots 74}{21\cdots 15}a^{23}+\frac{83\cdots 88}{64\cdots 45}a^{22}-\frac{35\cdots 76}{71\cdots 05}a^{21}+\frac{24\cdots 52}{21\cdots 15}a^{20}-\frac{27\cdots 21}{12\cdots 69}a^{19}+\frac{34\cdots 58}{64\cdots 45}a^{18}-\frac{21\cdots 29}{21\cdots 15}a^{17}+\frac{15\cdots 23}{14\cdots 41}a^{16}-\frac{29\cdots 96}{64\cdots 45}a^{15}+\frac{28\cdots 69}{64\cdots 45}a^{14}+\frac{58\cdots 31}{64\cdots 45}a^{13}+\frac{61\cdots 71}{12\cdots 69}a^{12}+\frac{22\cdots 15}{12\cdots 69}a^{11}-\frac{19\cdots 18}{64\cdots 45}a^{10}+\frac{42\cdots 02}{64\cdots 45}a^{9}-\frac{62\cdots 96}{21\cdots 15}a^{8}+\frac{83\cdots 44}{64\cdots 45}a^{7}+\frac{13\cdots 31}{64\cdots 45}a^{6}-\frac{22\cdots 81}{14\cdots 41}a^{5}-\frac{11\cdots 44}{71\cdots 05}a^{4}+\frac{13\cdots 01}{64\cdots 45}a^{3}+\frac{44\cdots 51}{12\cdots 69}a^{2}+\frac{16\cdots 57}{71\cdots 05}a+\frac{29\cdots 42}{71\cdots 05}$, $\frac{76\cdots 48}{21\cdots 15}a^{24}-\frac{22\cdots 14}{42\cdots 23}a^{23}-\frac{70\cdots 48}{14\cdots 41}a^{22}+\frac{36\cdots 11}{42\cdots 23}a^{21}-\frac{10\cdots 73}{21\cdots 15}a^{20}+\frac{46\cdots 37}{21\cdots 15}a^{19}-\frac{14\cdots 16}{42\cdots 23}a^{18}-\frac{32\cdots 14}{71\cdots 05}a^{17}+\frac{62\cdots 06}{21\cdots 15}a^{16}+\frac{84\cdots 54}{21\cdots 15}a^{15}+\frac{78\cdots 60}{42\cdots 23}a^{14}+\frac{55\cdots 22}{71\cdots 05}a^{13}+\frac{50\cdots 63}{42\cdots 23}a^{12}-\frac{65\cdots 81}{21\cdots 15}a^{11}-\frac{34\cdots 43}{71\cdots 05}a^{10}+\frac{43\cdots 37}{71\cdots 05}a^{9}-\frac{10\cdots 04}{71\cdots 05}a^{8}+\frac{42\cdots 02}{71\cdots 05}a^{7}-\frac{61\cdots 76}{21\cdots 15}a^{6}-\frac{82\cdots 82}{21\cdots 15}a^{5}+\frac{44\cdots 05}{14\cdots 41}a^{4}-\frac{73\cdots 28}{71\cdots 05}a^{3}+\frac{14\cdots 01}{21\cdots 15}a^{2}-\frac{28\cdots 83}{71\cdots 05}a-\frac{10\cdots 11}{14\cdots 41}$, $\frac{12\cdots 16}{42\cdots 23}a^{24}-\frac{22\cdots 37}{64\cdots 45}a^{23}-\frac{91\cdots 37}{21\cdots 15}a^{22}+\frac{78\cdots 02}{64\cdots 45}a^{21}-\frac{18\cdots 06}{71\cdots 05}a^{20}+\frac{13\cdots 61}{21\cdots 15}a^{19}-\frac{17\cdots 46}{12\cdots 69}a^{18}+\frac{13\cdots 86}{64\cdots 45}a^{17}-\frac{67\cdots 36}{42\cdots 23}a^{16}+\frac{14\cdots 21}{21\cdots 15}a^{15}+\frac{36\cdots 81}{64\cdots 45}a^{14}+\frac{63\cdots 37}{12\cdots 69}a^{13}+\frac{96\cdots 84}{64\cdots 45}a^{12}-\frac{24\cdots 98}{64\cdots 45}a^{11}+\frac{19\cdots 82}{64\cdots 45}a^{10}-\frac{20\cdots 91}{64\cdots 45}a^{9}+\frac{24\cdots 73}{64\cdots 45}a^{8}-\frac{30\cdots 83}{71\cdots 05}a^{7}-\frac{38\cdots 22}{64\cdots 45}a^{6}+\frac{37\cdots 58}{64\cdots 45}a^{5}-\frac{60\cdots 22}{71\cdots 05}a^{4}+\frac{14\cdots 83}{21\cdots 15}a^{3}-\frac{89\cdots 47}{64\cdots 45}a^{2}+\frac{10\cdots 88}{64\cdots 45}a-\frac{10\cdots 36}{71\cdots 05}$, $\frac{64\cdots 24}{64\cdots 45}a^{24}-\frac{52\cdots 61}{64\cdots 45}a^{23}-\frac{93\cdots 86}{64\cdots 45}a^{22}+\frac{42\cdots 77}{12\cdots 69}a^{21}-\frac{35\cdots 08}{71\cdots 05}a^{20}+\frac{85\cdots 39}{64\cdots 45}a^{19}-\frac{16\cdots 63}{64\cdots 45}a^{18}+\frac{14\cdots 79}{64\cdots 45}a^{17}+\frac{12\cdots 65}{42\cdots 23}a^{16}+\frac{93\cdots 22}{64\cdots 45}a^{15}+\frac{51\cdots 24}{12\cdots 69}a^{14}+\frac{20\cdots 86}{12\cdots 69}a^{13}+\frac{10\cdots 42}{21\cdots 15}a^{12}-\frac{65\cdots 01}{64\cdots 45}a^{11}-\frac{31\cdots 56}{21\cdots 15}a^{10}+\frac{49\cdots 46}{71\cdots 05}a^{9}-\frac{11\cdots 02}{64\cdots 45}a^{8}+\frac{17\cdots 97}{12\cdots 69}a^{7}-\frac{75\cdots 67}{71\cdots 05}a^{6}+\frac{14\cdots 58}{64\cdots 45}a^{5}+\frac{60\cdots 52}{21\cdots 15}a^{4}-\frac{96\cdots 54}{64\cdots 45}a^{3}+\frac{11\cdots 16}{64\cdots 45}a^{2}-\frac{18\cdots 11}{64\cdots 45}a+\frac{13\cdots 08}{14\cdots 41}$, $\frac{86\cdots 19}{64\cdots 45}a^{24}+\frac{20\cdots 13}{64\cdots 45}a^{23}-\frac{84\cdots 26}{42\cdots 23}a^{22}+\frac{45\cdots 29}{42\cdots 23}a^{21}-\frac{56\cdots 83}{21\cdots 15}a^{20}+\frac{37\cdots 09}{64\cdots 45}a^{19}-\frac{57\cdots 13}{42\cdots 23}a^{18}+\frac{52\cdots 27}{21\cdots 15}a^{17}-\frac{61\cdots 71}{21\cdots 15}a^{16}+\frac{28\cdots 71}{64\cdots 45}a^{15}-\frac{29\cdots 61}{71\cdots 05}a^{14}-\frac{32\cdots 17}{64\cdots 45}a^{13}+\frac{54\cdots 64}{12\cdots 69}a^{12}-\frac{21\cdots 16}{64\cdots 45}a^{11}+\frac{42\cdots 42}{71\cdots 05}a^{10}-\frac{30\cdots 73}{64\cdots 45}a^{9}+\frac{24\cdots 57}{64\cdots 45}a^{8}-\frac{19\cdots 48}{64\cdots 45}a^{7}-\frac{21\cdots 68}{21\cdots 15}a^{6}+\frac{52\cdots 99}{21\cdots 15}a^{5}-\frac{27\cdots 41}{21\cdots 15}a^{4}+\frac{36\cdots 82}{64\cdots 45}a^{3}-\frac{59\cdots 91}{21\cdots 15}a^{2}-\frac{48\cdots 58}{71\cdots 05}a+\frac{57\cdots 77}{71\cdots 05}$, $\frac{74\cdots 86}{71\cdots 05}a^{24}+\frac{74\cdots 69}{12\cdots 69}a^{23}-\frac{33\cdots 91}{21\cdots 15}a^{22}+\frac{30\cdots 03}{64\cdots 45}a^{21}-\frac{42\cdots 57}{42\cdots 23}a^{20}+\frac{34\cdots 28}{14\cdots 41}a^{19}-\frac{66\cdots 31}{12\cdots 69}a^{18}+\frac{50\cdots 66}{64\cdots 45}a^{17}-\frac{41\cdots 67}{71\cdots 05}a^{16}+\frac{50\cdots 63}{21\cdots 15}a^{15}+\frac{12\cdots 09}{64\cdots 45}a^{14}+\frac{60\cdots 93}{64\cdots 45}a^{13}+\frac{33\cdots 36}{64\cdots 45}a^{12}-\frac{18\cdots 51}{12\cdots 69}a^{11}+\frac{17\cdots 55}{12\cdots 69}a^{10}-\frac{66\cdots 42}{64\cdots 45}a^{9}+\frac{76\cdots 99}{64\cdots 45}a^{8}-\frac{32\cdots 82}{71\cdots 05}a^{7}-\frac{35\cdots 23}{64\cdots 45}a^{6}+\frac{28\cdots 73}{64\cdots 45}a^{5}-\frac{53\cdots 01}{21\cdots 15}a^{4}+\frac{10\cdots 07}{71\cdots 05}a^{3}+\frac{12\cdots 19}{64\cdots 45}a^{2}+\frac{60\cdots 93}{64\cdots 45}a+\frac{34\cdots 77}{71\cdots 05}$ 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:  \( 6619342681.783736 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{10}\cdot 6619342681.783736 \cdot 1}{2\cdot\sqrt{29802322387695312500000000000000000000}}\cr\approx \mathstrut & 1.86040919371447 \end{aligned}\] (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 - 45*x^22 + 95*x^21 - 235*x^20 + 500*x^19 - 760*x^18 + 605*x^17 - 2330*x^16 - 1835*x^15 - 1375*x^14 - 5310*x^13 + 13755*x^12 - 12555*x^11 + 11755*x^10 - 13250*x^9 + 1465*x^8 + 3355*x^7 - 3655*x^6 + 3280*x^5 - 2125*x^4 + 215*x^3 - 295*x^2 - 55*x + 9) 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 - 45*x^22 + 95*x^21 - 235*x^20 + 500*x^19 - 760*x^18 + 605*x^17 - 2330*x^16 - 1835*x^15 - 1375*x^14 - 5310*x^13 + 13755*x^12 - 12555*x^11 + 11755*x^10 - 13250*x^9 + 1465*x^8 + 3355*x^7 - 3655*x^6 + 3280*x^5 - 2125*x^4 + 215*x^3 - 295*x^2 - 55*x + 9, 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 - 45*x^22 + 95*x^21 - 235*x^20 + 500*x^19 - 760*x^18 + 605*x^17 - 2330*x^16 - 1835*x^15 - 1375*x^14 - 5310*x^13 + 13755*x^12 - 12555*x^11 + 11755*x^10 - 13250*x^9 + 1465*x^8 + 3355*x^7 - 3655*x^6 + 3280*x^5 - 2125*x^4 + 215*x^3 - 295*x^2 - 55*x + 9); 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 - 45*x^22 + 95*x^21 - 235*x^20 + 500*x^19 - 760*x^18 + 605*x^17 - 2330*x^16 - 1835*x^15 - 1375*x^14 - 5310*x^13 + 13755*x^12 - 12555*x^11 + 11755*x^10 - 13250*x^9 + 1465*x^8 + 3355*x^7 - 3655*x^6 + 3280*x^5 - 2125*x^4 + 215*x^3 - 295*x^2 - 55*x + 9); 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 ${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.1.0.1}{1} }^{5}$ R $20{,}\,{\href{/padicField/7.5.0.1}{5} }$ ${\href{/padicField/11.5.0.1}{5} }^{5}$ ${\href{/padicField/13.4.0.1}{4} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{5}$ $20{,}\,{\href{/padicField/17.5.0.1}{5} }$ ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }$ $20{,}\,{\href{/padicField/23.5.0.1}{5} }$ ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.5.0.1}{5} }$ ${\href{/padicField/31.5.0.1}{5} }^{5}$ $20{,}\,{\href{/padicField/37.5.0.1}{5} }$ ${\href{/padicField/41.5.0.1}{5} }^{5}$ $20{,}\,{\href{/padicField/43.5.0.1}{5} }$ $20{,}\,{\href{/padicField/47.5.0.1}{5} }$ $20{,}\,{\href{/padicField/53.5.0.1}{5} }$ ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.5.0.1}{5} }$

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)