Properties

Label 24.4.121...125.5
Degree $24$
Signature $(4, 10)$
Discriminant $1.211\times 10^{72}$
Root discriminant \(1008.02\)
Ramified primes $5,109$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $\GL(2,5)$ (as 24T1353)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^24 + 16350*x^20 - 1814850*x^18 + 5986825*x^16 + 3043866965*x^14 - 37017062175*x^12 - 1856266269475*x^10 + 38406857938650*x^8 + 183163137913575*x^6 - 7199514724704670*x^4 + 63149113723696375*x^2 + 613116592600078125)
 
Copy content gp:K = bnfinit(y^24 + 16350*y^20 - 1814850*y^18 + 5986825*y^16 + 3043866965*y^14 - 37017062175*y^12 - 1856266269475*y^10 + 38406857938650*y^8 + 183163137913575*y^6 - 7199514724704670*y^4 + 63149113723696375*y^2 + 613116592600078125, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 16350*x^20 - 1814850*x^18 + 5986825*x^16 + 3043866965*x^14 - 37017062175*x^12 - 1856266269475*x^10 + 38406857938650*x^8 + 183163137913575*x^6 - 7199514724704670*x^4 + 63149113723696375*x^2 + 613116592600078125);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 + 16350*x^20 - 1814850*x^18 + 5986825*x^16 + 3043866965*x^14 - 37017062175*x^12 - 1856266269475*x^10 + 38406857938650*x^8 + 183163137913575*x^6 - 7199514724704670*x^4 + 63149113723696375*x^2 + 613116592600078125)
 

\( x^{24} + 16350 x^{20} - 1814850 x^{18} + 5986825 x^{16} + 3043866965 x^{14} - 37017062175 x^{12} + \cdots + 61\!\cdots\!25 \) 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:  $24$
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:  $(4, 10)$
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:   \(1211192363043189952963494621037336955654250561565277166664600372314453125\) \(\medspace = 5^{39}\cdot 109^{22}\) 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:  \(1008.02\)
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:  $5^{39/20}109^{19/20}\approx 1988.5965494204843$
Ramified primes:   \(5\), \(109\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{4}$, $\frac{1}{1090}a^{20}-\frac{1}{2}a^{5}$, $\frac{1}{5450}a^{21}+\frac{1}{5}a^{11}-\frac{1}{2}a^{6}+\frac{2}{5}a$, $\frac{1}{32\cdots 00}a^{22}+\frac{58\cdots 69}{12\cdots 80}a^{20}+\frac{16\cdots 11}{11\cdots 20}a^{18}+\frac{11\cdots 89}{11\cdots 20}a^{16}-\frac{66\cdots 89}{59\cdots 60}a^{14}-\frac{52\cdots 73}{59\cdots 00}a^{12}+\frac{10\cdots 63}{29\cdots 80}a^{10}-\frac{51\cdots 31}{11\cdots 20}a^{8}-\frac{55\cdots 01}{11\cdots 20}a^{6}-\frac{1}{2}a^{5}-\frac{23\cdots 59}{59\cdots 60}a^{4}-\frac{27\cdots 71}{10\cdots 00}a^{2}-\frac{38\cdots 81}{23\cdots 64}$, $\frac{1}{41\cdots 00}a^{23}-\frac{1}{64\cdots 00}a^{22}-\frac{13\cdots 47}{25\cdots 60}a^{21}+\frac{59\cdots 63}{25\cdots 60}a^{20}-\frac{94\cdots 23}{50\cdots 00}a^{19}-\frac{16\cdots 11}{23\cdots 40}a^{18}+\frac{12\cdots 03}{50\cdots 00}a^{17}-\frac{11\cdots 89}{23\cdots 40}a^{16}+\frac{30\cdots 47}{58\cdots 00}a^{15}+\frac{66\cdots 89}{11\cdots 20}a^{14}-\frac{15\cdots 73}{76\cdots 00}a^{13}+\frac{52\cdots 73}{11\cdots 00}a^{12}-\frac{16\cdots 39}{12\cdots 00}a^{11}+\frac{19\cdots 17}{59\cdots 60}a^{10}+\frac{41\cdots 69}{15\cdots 00}a^{9}-\frac{67\cdots 89}{23\cdots 40}a^{8}+\frac{11\cdots 73}{50\cdots 00}a^{7}-\frac{62\cdots 19}{23\cdots 40}a^{6}+\frac{38\cdots 67}{25\cdots 00}a^{5}-\frac{35\cdots 01}{11\cdots 20}a^{4}+\frac{53\cdots 29}{13\cdots 00}a^{3}+\frac{27\cdots 71}{21\cdots 00}a^{2}-\frac{25\cdots 81}{30\cdots 00}a-\frac{19\cdots 83}{47\cdots 28}$ 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:  $2$, $3$

Class group and class number

Ideal class group:  $C_{4}$, which has order $4$ (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:  $C_{4}\times C_{2}$, which has order $8$ (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:  $13$
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:   $\frac{17\cdots 47}{16\cdots 10}a^{22}-\frac{20\cdots 27}{32\cdots 22}a^{20}+\frac{52\cdots 13}{29\cdots 58}a^{18}-\frac{61\cdots 05}{29\cdots 58}a^{16}+\frac{27\cdots 22}{14\cdots 79}a^{14}+\frac{94\cdots 39}{29\cdots 58}a^{12}-\frac{87\cdots 49}{14\cdots 79}a^{10}-\frac{49\cdots 05}{29\cdots 58}a^{8}+\frac{15\cdots 59}{29\cdots 58}a^{6}-\frac{16\cdots 76}{14\cdots 79}a^{4}-\frac{76\cdots 20}{10\cdots 61}a^{2}+\frac{32\cdots 67}{29\cdots 58}$, $\frac{15\cdots 52}{40\cdots 75}a^{22}+\frac{34\cdots 08}{14\cdots 79}a^{20}-\frac{93\cdots 39}{14\cdots 79}a^{18}+\frac{10\cdots 53}{14\cdots 79}a^{16}-\frac{99\cdots 18}{14\cdots 79}a^{14}-\frac{83\cdots 89}{74\cdots 95}a^{12}+\frac{31\cdots 32}{14\cdots 79}a^{10}+\frac{87\cdots 15}{14\cdots 79}a^{8}-\frac{27\cdots 56}{14\cdots 79}a^{6}+\frac{57\cdots 33}{14\cdots 79}a^{4}+\frac{13\cdots 23}{53\cdots 05}a^{2}-\frac{58\cdots 49}{14\cdots 79}$, $\frac{21\cdots 41}{80\cdots 50}a^{22}-\frac{18\cdots 61}{14\cdots 90}a^{20}-\frac{14\cdots 47}{29\cdots 58}a^{18}+\frac{76\cdots 49}{29\cdots 58}a^{16}+\frac{15\cdots 48}{14\cdots 79}a^{14}-\frac{50\cdots 17}{14\cdots 90}a^{12}-\frac{86\cdots 39}{14\cdots 79}a^{10}+\frac{68\cdots 55}{29\cdots 58}a^{8}+\frac{11\cdots 13}{29\cdots 58}a^{6}-\frac{48\cdots 35}{14\cdots 79}a^{4}+\frac{23\cdots 52}{53\cdots 05}a^{2}+\frac{10\cdots 07}{29\cdots 58}$, $\frac{86\cdots 62}{66\cdots 81}a^{22}-\frac{16\cdots 02}{66\cdots 81}a^{20}-\frac{11\cdots 37}{60\cdots 09}a^{18}-\frac{30\cdots 42}{60\cdots 09}a^{16}+\frac{52\cdots 28}{60\cdots 09}a^{14}+\frac{15\cdots 67}{60\cdots 09}a^{12}+\frac{56\cdots 32}{60\cdots 09}a^{10}-\frac{22\cdots 94}{60\cdots 09}a^{8}-\frac{47\cdots 89}{60\cdots 09}a^{6}+\frac{18\cdots 48}{60\cdots 09}a^{4}-\frac{41\cdots 00}{43\cdots 31}a^{2}-\frac{39\cdots 26}{60\cdots 09}$, $\frac{99\cdots 43}{80\cdots 50}a^{22}+\frac{21\cdots 61}{29\cdots 58}a^{20}-\frac{60\cdots 99}{29\cdots 58}a^{18}+\frac{70\cdots 69}{29\cdots 58}a^{16}-\frac{31\cdots 55}{14\cdots 79}a^{14}-\frac{53\cdots 81}{14\cdots 90}a^{12}+\frac{99\cdots 51}{14\cdots 79}a^{10}+\frac{56\cdots 07}{29\cdots 58}a^{8}-\frac{17\cdots 49}{29\cdots 58}a^{6}+\frac{18\cdots 84}{14\cdots 79}a^{4}+\frac{43\cdots 61}{53\cdots 05}a^{2}-\frac{37\cdots 13}{29\cdots 58}$, $\frac{24\cdots 56}{20\cdots 75}a^{22}-\frac{45\cdots 16}{16\cdots 11}a^{20}+\frac{16\cdots 79}{74\cdots 95}a^{18}+\frac{65\cdots 51}{74\cdots 95}a^{16}-\frac{55\cdots 92}{74\cdots 95}a^{14}-\frac{37\cdots 37}{37\cdots 75}a^{12}+\frac{50\cdots 93}{74\cdots 95}a^{10}-\frac{70\cdots 84}{74\cdots 95}a^{8}-\frac{67\cdots 29}{74\cdots 95}a^{6}+\frac{17\cdots 73}{74\cdots 95}a^{4}-\frac{42\cdots 46}{26\cdots 25}a^{2}-\frac{26\cdots 47}{14\cdots 79}$, $\frac{13\cdots 49}{39\cdots 00}a^{23}-\frac{18\cdots 17}{80\cdots 00}a^{22}+\frac{51\cdots 53}{32\cdots 20}a^{21}-\frac{70\cdots 19}{64\cdots 44}a^{20}+\frac{31\cdots 23}{48\cdots 00}a^{19}-\frac{12\cdots 77}{29\cdots 80}a^{18}-\frac{16\cdots 03}{48\cdots 00}a^{17}+\frac{66\cdots 87}{29\cdots 80}a^{16}-\frac{96\cdots 11}{73\cdots 50}a^{15}+\frac{13\cdots 63}{14\cdots 90}a^{14}+\frac{31\cdots 73}{73\cdots 00}a^{13}-\frac{43\cdots 09}{14\cdots 00}a^{12}+\frac{90\cdots 89}{12\cdots 25}a^{11}-\frac{37\cdots 21}{74\cdots 95}a^{10}-\frac{43\cdots 69}{14\cdots 00}a^{9}+\frac{59\cdots 87}{29\cdots 80}a^{8}-\frac{23\cdots 23}{48\cdots 00}a^{7}+\frac{95\cdots 57}{29\cdots 80}a^{6}+\frac{10\cdots 83}{24\cdots 50}a^{5}-\frac{41\cdots 87}{14\cdots 90}a^{4}-\frac{74\cdots 04}{13\cdots 25}a^{3}+\frac{10\cdots 32}{26\cdots 25}a^{2}-\frac{13\cdots 69}{29\cdots 00}a+\frac{18\cdots 35}{59\cdots 16}$, $\frac{42\cdots 49}{25\cdots 75}a^{23}-\frac{94\cdots 67}{80\cdots 00}a^{22}-\frac{33\cdots 33}{40\cdots 75}a^{21}-\frac{19\cdots 43}{32\cdots 20}a^{20}-\frac{19\cdots 71}{63\cdots 50}a^{19}-\frac{65\cdots 87}{29\cdots 80}a^{18}+\frac{45\cdots 78}{31\cdots 25}a^{17}+\frac{29\cdots 17}{29\cdots 80}a^{16}+\frac{90\cdots 63}{14\cdots 50}a^{15}+\frac{32\cdots 44}{74\cdots 95}a^{14}-\frac{18\cdots 21}{95\cdots 50}a^{13}-\frac{20\cdots 09}{14\cdots 00}a^{12}-\frac{22\cdots 57}{63\cdots 50}a^{11}-\frac{18\cdots 31}{74\cdots 95}a^{10}+\frac{24\cdots 13}{19\cdots 50}a^{9}+\frac{27\cdots 47}{29\cdots 80}a^{8}+\frac{72\cdots 73}{31\cdots 25}a^{7}+\frac{48\cdots 07}{29\cdots 80}a^{6}-\frac{11\cdots 07}{63\cdots 50}a^{5}-\frac{98\cdots 56}{74\cdots 95}a^{4}+\frac{16\cdots 57}{68\cdots 50}a^{3}+\frac{46\cdots 32}{26\cdots 25}a^{2}+\frac{76\cdots 89}{38\cdots 90}a+\frac{84\cdots 67}{59\cdots 16}$, $\frac{21\cdots 11}{69\cdots 00}a^{23}+\frac{71\cdots 67}{32\cdots 00}a^{22}-\frac{10\cdots 21}{64\cdots 00}a^{21}+\frac{28\cdots 83}{25\cdots 76}a^{20}-\frac{14\cdots 41}{25\cdots 00}a^{19}+\frac{49\cdots 77}{11\cdots 20}a^{18}+\frac{68\cdots 01}{25\cdots 00}a^{17}-\frac{22\cdots 37}{11\cdots 20}a^{16}+\frac{11\cdots 83}{97\cdots 00}a^{15}-\frac{49\cdots 83}{59\cdots 60}a^{14}-\frac{45\cdots 97}{12\cdots 00}a^{13}+\frac{15\cdots 09}{59\cdots 00}a^{12}-\frac{41\cdots 13}{63\cdots 00}a^{11}+\frac{13\cdots 81}{29\cdots 80}a^{10}+\frac{62\cdots 41}{25\cdots 00}a^{9}-\frac{20\cdots 77}{11\cdots 20}a^{8}+\frac{10\cdots 91}{25\cdots 00}a^{7}-\frac{36\cdots 07}{11\cdots 20}a^{6}-\frac{44\cdots 11}{12\cdots 00}a^{5}+\frac{14\cdots 07}{59\cdots 60}a^{4}+\frac{10\cdots 31}{22\cdots 00}a^{3}-\frac{35\cdots 57}{10\cdots 00}a^{2}+\frac{19\cdots 51}{50\cdots 00}a-\frac{63\cdots 83}{23\cdots 64}$, $\frac{94\cdots 21}{69\cdots 00}a^{23}+\frac{26\cdots 57}{32\cdots 00}a^{22}+\frac{33\cdots 11}{64\cdots 00}a^{21}+\frac{88\cdots 61}{25\cdots 76}a^{20}+\frac{62\cdots 51}{25\cdots 00}a^{19}+\frac{17\cdots 47}{11\cdots 20}a^{18}-\frac{38\cdots 11}{25\cdots 00}a^{17}-\frac{98\cdots 47}{11\cdots 20}a^{16}-\frac{40\cdots 13}{97\cdots 00}a^{15}-\frac{16\cdots 93}{59\cdots 60}a^{14}+\frac{28\cdots 67}{12\cdots 00}a^{13}+\frac{68\cdots 39}{59\cdots 00}a^{12}+\frac{96\cdots 93}{63\cdots 00}a^{11}+\frac{35\cdots 21}{29\cdots 80}a^{10}-\frac{40\cdots 51}{25\cdots 00}a^{9}-\frac{93\cdots 07}{11\cdots 20}a^{8}+\frac{19\cdots 99}{25\cdots 00}a^{7}+\frac{30\cdots 43}{11\cdots 20}a^{6}+\frac{37\cdots 21}{12\cdots 00}a^{5}+\frac{73\cdots 57}{59\cdots 60}a^{4}-\frac{83\cdots 41}{22\cdots 00}a^{3}-\frac{22\cdots 97}{10\cdots 00}a^{2}-\frac{14\cdots 61}{50\cdots 00}a-\frac{36\cdots 13}{23\cdots 64}$, $\frac{44\cdots 67}{69\cdots 00}a^{23}+\frac{14\cdots 67}{32\cdots 00}a^{22}+\frac{19\cdots 37}{64\cdots 00}a^{21}+\frac{26\cdots 83}{12\cdots 80}a^{20}+\frac{30\cdots 77}{25\cdots 00}a^{19}+\frac{97\cdots 57}{11\cdots 20}a^{18}-\frac{15\cdots 97}{25\cdots 00}a^{17}-\frac{49\cdots 57}{11\cdots 20}a^{16}-\frac{24\cdots 51}{97\cdots 00}a^{15}-\frac{10\cdots 43}{59\cdots 60}a^{14}+\frac{10\cdots 09}{12\cdots 00}a^{13}+\frac{32\cdots 09}{59\cdots 00}a^{12}+\frac{90\cdots 11}{63\cdots 00}a^{11}+\frac{28\cdots 21}{29\cdots 80}a^{10}-\frac{13\cdots 77}{25\cdots 00}a^{9}-\frac{44\cdots 77}{11\cdots 20}a^{8}-\frac{27\cdots 27}{25\cdots 00}a^{7}-\frac{79\cdots 47}{11\cdots 20}a^{6}+\frac{95\cdots 67}{12\cdots 00}a^{5}+\frac{31\cdots 47}{59\cdots 60}a^{4}-\frac{23\cdots 57}{22\cdots 00}a^{3}-\frac{75\cdots 57}{10\cdots 00}a^{2}-\frac{41\cdots 47}{50\cdots 00}a-\frac{13\cdots 19}{23\cdots 64}$, $\frac{49\cdots 61}{51\cdots 00}a^{23}+\frac{89\cdots 46}{20\cdots 75}a^{22}-\frac{12\cdots 69}{64\cdots 44}a^{21}+\frac{60\cdots 32}{80\cdots 55}a^{20}-\frac{10\cdots 47}{63\cdots 00}a^{19}+\frac{54\cdots 36}{74\cdots 95}a^{18}+\frac{90\cdots 67}{63\cdots 00}a^{17}-\frac{10\cdots 27}{14\cdots 90}a^{16}+\frac{91\cdots 29}{36\cdots 75}a^{15}-\frac{15\cdots 01}{14\cdots 90}a^{14}-\frac{24\cdots 97}{95\cdots 00}a^{13}+\frac{46\cdots 92}{37\cdots 75}a^{12}-\frac{35\cdots 46}{15\cdots 25}a^{11}+\frac{12\cdots 59}{14\cdots 90}a^{10}+\frac{28\cdots 91}{19\cdots 00}a^{9}-\frac{56\cdots 56}{74\cdots 95}a^{8}+\frac{66\cdots 47}{63\cdots 00}a^{7}-\frac{40\cdots 87}{14\cdots 90}a^{6}-\frac{28\cdots 06}{15\cdots 25}a^{5}+\frac{61\cdots 97}{74\cdots 95}a^{4}+\frac{11\cdots 37}{34\cdots 50}a^{3}-\frac{90\cdots 53}{53\cdots 50}a^{2}+\frac{90\cdots 41}{38\cdots 00}a-\frac{35\cdots 95}{29\cdots 58}$, $\frac{10\cdots 03}{20\cdots 00}a^{23}+\frac{14\cdots 46}{20\cdots 75}a^{22}-\frac{10\cdots 97}{32\cdots 00}a^{21}-\frac{17\cdots 43}{80\cdots 55}a^{20}+\frac{20\cdots 31}{25\cdots 00}a^{19}+\frac{91\cdots 86}{74\cdots 95}a^{18}-\frac{24\cdots 91}{25\cdots 00}a^{17}-\frac{24\cdots 37}{14\cdots 90}a^{16}+\frac{11\cdots 91}{29\cdots 00}a^{15}+\frac{81\cdots 39}{14\cdots 90}a^{14}+\frac{90\cdots 81}{38\cdots 00}a^{13}+\frac{24\cdots 92}{37\cdots 75}a^{12}-\frac{50\cdots 77}{63\cdots 50}a^{11}-\frac{84\cdots 51}{14\cdots 90}a^{10}+\frac{35\cdots 07}{76\cdots 00}a^{9}+\frac{46\cdots 79}{74\cdots 95}a^{8}+\frac{41\cdots 19}{25\cdots 00}a^{7}+\frac{12\cdots 33}{14\cdots 90}a^{6}-\frac{32\cdots 49}{12\cdots 00}a^{5}-\frac{13\cdots 48}{74\cdots 95}a^{4}+\frac{36\cdots 56}{34\cdots 25}a^{3}+\frac{53\cdots 97}{53\cdots 50}a^{2}+\frac{27\cdots 69}{15\cdots 60}a+\frac{38\cdots 93}{29\cdots 58}$ 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:  \( 57801913481412370000000000000 \) (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:  \( 3 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 57801913481412370000000000000 \cdot 4}{2\cdot\sqrt{1211192363043189952963494621037336955654250561565277166664600372314453125}}\cr\approx \mathstrut & 161.170040704675 \end{aligned}\] (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^24 + 16350*x^20 - 1814850*x^18 + 5986825*x^16 + 3043866965*x^14 - 37017062175*x^12 - 1856266269475*x^10 + 38406857938650*x^8 + 183163137913575*x^6 - 7199514724704670*x^4 + 63149113723696375*x^2 + 613116592600078125) 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^24 + 16350*x^20 - 1814850*x^18 + 5986825*x^16 + 3043866965*x^14 - 37017062175*x^12 - 1856266269475*x^10 + 38406857938650*x^8 + 183163137913575*x^6 - 7199514724704670*x^4 + 63149113723696375*x^2 + 613116592600078125, 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^24 + 16350*x^20 - 1814850*x^18 + 5986825*x^16 + 3043866965*x^14 - 37017062175*x^12 - 1856266269475*x^10 + 38406857938650*x^8 + 183163137913575*x^6 - 7199514724704670*x^4 + 63149113723696375*x^2 + 613116592600078125); 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^24 + 16350*x^20 - 1814850*x^18 + 5986825*x^16 + 3043866965*x^14 - 37017062175*x^12 - 1856266269475*x^10 + 38406857938650*x^8 + 183163137913575*x^6 - 7199514724704670*x^4 + 63149113723696375*x^2 + 613116592600078125); 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

$\GL(2,5)$ (as 24T1353):

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 non-solvable group of order 480
The 24 conjugacy class representatives for $\GL(2,5)$
Character table for $\GL(2,5)$

Intermediate fields

6.2.275699533203125.2, 12.4.4515387868103250949878692626953125.1

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 24 siblings: 24.4.30279809076079748824087365525933423891356264039131929166615009307861328125.5, 24.4.30279809076079748824087365525933423891356264039131929166615009307861328125.7
Arithmetically equivalent sibling: 24.4.1211192363043189952963494621037336955654250561565277166664600372314453125.1
Minimal sibling: 24.4.1211192363043189952963494621037336955654250561565277166664600372314453125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{5}{,}\,{\href{/padicField/2.1.0.1}{1} }^{4}$ ${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ R $24$ $20{,}\,{\href{/padicField/11.4.0.1}{4} }$ ${\href{/padicField/13.4.0.1}{4} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.5.0.1}{5} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ $24$ ${\href{/padicField/29.2.0.1}{2} }^{10}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ $24$ ${\href{/padicField/41.12.0.1}{12} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ $24$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
5.1.20.39a1.3$x^{20} + 25 x^{4} + 5$$20$$1$$39$20T5$$[\frac{9}{4}]_{4}$$
\(109\) Copy content Toggle raw display 109.1.4.3a1.1$x^{4} + 109$$4$$1$$3$$C_4$$$[\ ]_{4}$$
109.1.20.19a1.3$x^{20} + 3924$$20$$1$$19$20T6$$[\ ]_{20}^{2}$$

Spectrum of ring of integers

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