Properties

Label 24.4.543...125.3
Degree $24$
Signature $(4, 10)$
Discriminant $5.437\times 10^{55}$
Root discriminant \(210.04\)
Ramified primes $5,43$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $\GL(2,5)$ (as 24T1353)

Related objects

Downloads

Learn more

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 - 12*x^23 + 69*x^22 - 253*x^21 + 261*x^20 + 2703*x^19 - 15406*x^18 + 39987*x^17 + 272986*x^16 - 2610282*x^15 + 6157629*x^14 + 3326657*x^13 - 48975994*x^12 + 129522333*x^11 - 171855021*x^10 + 68435337*x^9 + 134996941*x^8 - 232660062*x^7 + 184844654*x^6 - 129342673*x^5 + 113722606*x^4 - 86398607*x^3 + 43132904*x^2 - 12596758*x + 1680671)
 
Copy content gp:K = bnfinit(y^24 - 12*y^23 + 69*y^22 - 253*y^21 + 261*y^20 + 2703*y^19 - 15406*y^18 + 39987*y^17 + 272986*y^16 - 2610282*y^15 + 6157629*y^14 + 3326657*y^13 - 48975994*y^12 + 129522333*y^11 - 171855021*y^10 + 68435337*y^9 + 134996941*y^8 - 232660062*y^7 + 184844654*y^6 - 129342673*y^5 + 113722606*y^4 - 86398607*y^3 + 43132904*y^2 - 12596758*y + 1680671, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 12*x^23 + 69*x^22 - 253*x^21 + 261*x^20 + 2703*x^19 - 15406*x^18 + 39987*x^17 + 272986*x^16 - 2610282*x^15 + 6157629*x^14 + 3326657*x^13 - 48975994*x^12 + 129522333*x^11 - 171855021*x^10 + 68435337*x^9 + 134996941*x^8 - 232660062*x^7 + 184844654*x^6 - 129342673*x^5 + 113722606*x^4 - 86398607*x^3 + 43132904*x^2 - 12596758*x + 1680671);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 - 12*x^23 + 69*x^22 - 253*x^21 + 261*x^20 + 2703*x^19 - 15406*x^18 + 39987*x^17 + 272986*x^16 - 2610282*x^15 + 6157629*x^14 + 3326657*x^13 - 48975994*x^12 + 129522333*x^11 - 171855021*x^10 + 68435337*x^9 + 134996941*x^8 - 232660062*x^7 + 184844654*x^6 - 129342673*x^5 + 113722606*x^4 - 86398607*x^3 + 43132904*x^2 - 12596758*x + 1680671)
 

\( x^{24} - 12 x^{23} + 69 x^{22} - 253 x^{21} + 261 x^{20} + 2703 x^{19} - 15406 x^{18} + 39987 x^{17} + \cdots + 1680671 \) 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:   \(54372494083060640194576311797602218575775623321533203125\) \(\medspace = 5^{33}\cdot 43^{20}\) 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:  \(210.04\)
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:  $5^{31/20}43^{9/10}\approx 357.7050872928245$
Ramified primes:   \(5\), \(43\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{86}a^{10}-\frac{5}{86}a^{9}-\frac{21}{86}a^{8}+\frac{14}{43}a^{7}-\frac{3}{86}a^{6}+\frac{19}{86}a^{5}-\frac{33}{86}a^{4}+\frac{17}{43}a^{3}-\frac{1}{86}a^{2}-\frac{19}{86}a+\frac{8}{43}$, $\frac{1}{86}a^{11}-\frac{3}{86}a^{9}+\frac{9}{86}a^{8}+\frac{4}{43}a^{7}-\frac{39}{86}a^{6}-\frac{12}{43}a^{5}-\frac{1}{43}a^{4}-\frac{3}{86}a^{3}-\frac{12}{43}a^{2}-\frac{18}{43}a-\frac{3}{43}$, $\frac{1}{86}a^{12}-\frac{3}{43}a^{9}-\frac{6}{43}a^{8}-\frac{41}{86}a^{7}+\frac{5}{43}a^{6}+\frac{6}{43}a^{5}-\frac{8}{43}a^{4}+\frac{35}{86}a^{3}-\frac{39}{86}a^{2}+\frac{23}{86}a+\frac{5}{86}$, $\frac{1}{86}a^{13}+\frac{1}{86}a^{9}+\frac{5}{86}a^{8}-\frac{37}{86}a^{7}+\frac{37}{86}a^{6}+\frac{6}{43}a^{5}-\frac{17}{43}a^{4}-\frac{7}{86}a^{3}+\frac{17}{86}a^{2}+\frac{10}{43}a+\frac{5}{43}$, $\frac{1}{3698}a^{14}-\frac{7}{3698}a^{13}+\frac{6}{1849}a^{12}+\frac{19}{3698}a^{11}-\frac{5}{1849}a^{10}-\frac{291}{3698}a^{9}-\frac{337}{1849}a^{8}+\frac{1239}{3698}a^{7}+\frac{335}{1849}a^{6}+\frac{1167}{3698}a^{5}-\frac{883}{3698}a^{4}-\frac{977}{3698}a^{3}-\frac{678}{1849}a^{2}+\frac{545}{1849}a-\frac{365}{1849}$, $\frac{1}{3698}a^{15}+\frac{3}{1849}a^{13}+\frac{17}{3698}a^{12}-\frac{3}{1849}a^{11}-\frac{17}{3698}a^{10}+\frac{213}{3698}a^{9}+\frac{477}{3698}a^{8}+\frac{694}{1849}a^{7}-\frac{507}{3698}a^{6}+\frac{805}{1849}a^{5}+\frac{76}{1849}a^{4}+\frac{577}{3698}a^{3}-\frac{1565}{3698}a^{2}+\frac{96}{1849}a+\frac{584}{1849}$, $\frac{1}{7396}a^{16}+\frac{4}{1849}a^{13}-\frac{35}{7396}a^{12}-\frac{1}{3698}a^{11}+\frac{15}{7396}a^{10}+\frac{244}{1849}a^{9}+\frac{93}{3698}a^{8}+\frac{584}{1849}a^{7}-\frac{2281}{7396}a^{6}+\frac{1821}{3698}a^{5}+\frac{981}{3698}a^{4}-\frac{764}{1849}a^{3}+\frac{1541}{3698}a^{2}-\frac{784}{1849}a+\frac{2961}{7396}$, $\frac{1}{318028}a^{17}+\frac{13}{318028}a^{16}+\frac{17}{159014}a^{15}-\frac{21}{159014}a^{14}-\frac{1367}{318028}a^{13}+\frac{457}{318028}a^{12}-\frac{285}{318028}a^{11}+\frac{571}{318028}a^{10}-\frac{11603}{159014}a^{9}+\frac{12789}{159014}a^{8}-\frac{4925}{318028}a^{7}+\frac{21005}{318028}a^{6}+\frac{72877}{159014}a^{5}-\frac{60215}{159014}a^{4}+\frac{28075}{159014}a^{3}-\frac{15196}{79507}a^{2}-\frac{13745}{318028}a-\frac{55841}{318028}$, $\frac{1}{318028}a^{18}-\frac{3}{159014}a^{16}+\frac{8}{79507}a^{15}+\frac{39}{318028}a^{14}-\frac{561}{159014}a^{13}+\frac{955}{318028}a^{12}-\frac{307}{79507}a^{11}-\frac{845}{159014}a^{10}+\frac{6635}{159014}a^{9}+\frac{197}{318028}a^{8}-\frac{21641}{159014}a^{7}+\frac{10237}{79507}a^{6}+\frac{38035}{79507}a^{5}+\frac{9104}{79507}a^{4}-\frac{33402}{79507}a^{3}-\frac{108149}{318028}a^{2}-\frac{60827}{159014}a-\frac{16822}{79507}$, $\frac{1}{318028}a^{19}-\frac{19}{318028}a^{16}-\frac{15}{318028}a^{15}+\frac{1}{159014}a^{14}+\frac{1697}{318028}a^{13}-\frac{335}{318028}a^{12}-\frac{334}{79507}a^{11}+\frac{1689}{318028}a^{10}-\frac{13995}{318028}a^{9}+\frac{2069}{79507}a^{8}-\frac{55877}{159014}a^{7}+\frac{101569}{318028}a^{6}-\frac{79149}{159014}a^{5}+\frac{27491}{159014}a^{4}+\frac{28371}{318028}a^{3}+\frac{6651}{159014}a^{2}+\frac{5625}{79507}a-\frac{48107}{318028}$, $\frac{1}{12025906021580}a^{20}-\frac{1}{1202590602158}a^{19}-\frac{2956219}{2405181204316}a^{18}-\frac{911290}{601295301079}a^{17}+\frac{3144885}{1202590602158}a^{16}-\frac{688121341}{6012953010790}a^{15}+\frac{56904861}{601295301079}a^{14}+\frac{1184199624}{601295301079}a^{13}+\frac{3053047623}{601295301079}a^{12}-\frac{2750067609}{601295301079}a^{11}-\frac{2250965813}{546632091890}a^{10}+\frac{124582569931}{1202590602158}a^{9}-\frac{444779058263}{2405181204316}a^{8}+\frac{144230822169}{601295301079}a^{7}-\frac{1009124130149}{2405181204316}a^{6}+\frac{1299139532478}{3006476505395}a^{5}+\frac{124615367739}{2405181204316}a^{4}-\frac{182674163084}{601295301079}a^{3}+\frac{11107655971}{218652836756}a^{2}+\frac{504697844619}{1202590602158}a-\frac{3000432051469}{12025906021580}$, $\frac{1}{12025906021580}a^{21}-\frac{268749}{218652836756}a^{19}-\frac{134371}{109326418378}a^{18}+\frac{44679}{1202590602158}a^{17}-\frac{1481301}{6012953010790}a^{16}+\frac{61056515}{1202590602158}a^{15}-\frac{116083295}{1202590602158}a^{14}+\frac{2575247550}{601295301079}a^{13}-\frac{1215355077}{601295301079}a^{12}+\frac{5315846137}{6012953010790}a^{11}+\frac{2238868822}{601295301079}a^{10}-\frac{452057295571}{2405181204316}a^{9}-\frac{104986974315}{601295301079}a^{8}-\frac{17204229131}{2405181204316}a^{7}+\frac{984597015663}{3006476505395}a^{6}+\frac{313295012245}{2405181204316}a^{5}-\frac{105056836345}{601295301079}a^{4}+\frac{494043123895}{2405181204316}a^{3}-\frac{509845473389}{1202590602158}a^{2}-\frac{3127347706939}{12025906021580}a+\frac{161893903542}{601295301079}$, $\frac{1}{18\cdots 00}a^{22}-\frac{1}{16\cdots 00}a^{21}+\frac{6288637989}{18\cdots 00}a^{20}-\frac{12577275901}{36\cdots 20}a^{19}-\frac{37\cdots 47}{36\cdots 20}a^{18}+\frac{29\cdots 19}{90\cdots 50}a^{17}+\frac{35\cdots 11}{90\cdots 50}a^{16}+\frac{52\cdots 31}{90\cdots 50}a^{15}-\frac{45\cdots 69}{36\cdots 20}a^{14}-\frac{26\cdots 09}{90\cdots 55}a^{13}-\frac{64\cdots 71}{18\cdots 00}a^{12}-\frac{12\cdots 86}{45\cdots 75}a^{11}+\frac{21\cdots 11}{18\cdots 00}a^{10}+\frac{11\cdots 91}{36\cdots 20}a^{9}+\frac{10\cdots 17}{18\cdots 11}a^{8}+\frac{15\cdots 17}{18\cdots 00}a^{7}-\frac{78\cdots 67}{18\cdots 00}a^{6}-\frac{94\cdots 77}{18\cdots 00}a^{5}-\frac{30\cdots 27}{72\cdots 44}a^{4}-\frac{65\cdots 27}{36\cdots 20}a^{3}+\frac{19\cdots 73}{90\cdots 50}a^{2}+\frac{24\cdots 79}{18\cdots 00}a-\frac{14\cdots 04}{45\cdots 75}$, $\frac{1}{18\cdots 00}a^{23}+\frac{1572159467}{45\cdots 75}a^{21}+\frac{3144319187}{90\cdots 50}a^{20}-\frac{18\cdots 79}{18\cdots 10}a^{19}+\frac{18\cdots 53}{18\cdots 00}a^{18}-\frac{23\cdots 97}{36\cdots 20}a^{17}+\frac{37\cdots 03}{15\cdots 50}a^{16}+\frac{11\cdots 87}{18\cdots 00}a^{15}+\frac{81\cdots 91}{72\cdots 44}a^{14}-\frac{19\cdots 19}{45\cdots 75}a^{13}+\frac{34\cdots 93}{72\cdots 44}a^{12}+\frac{49\cdots 51}{90\cdots 50}a^{11}-\frac{39\cdots 37}{90\cdots 50}a^{10}-\frac{37\cdots 09}{36\cdots 20}a^{9}+\frac{45\cdots 67}{18\cdots 00}a^{8}-\frac{14\cdots 31}{36\cdots 20}a^{7}+\frac{15\cdots 09}{45\cdots 75}a^{6}+\frac{21\cdots 57}{45\cdots 75}a^{5}+\frac{40\cdots 72}{90\cdots 55}a^{4}+\frac{24\cdots 61}{18\cdots 00}a^{3}+\frac{12\cdots 07}{36\cdots 20}a^{2}+\frac{33\cdots 89}{90\cdots 50}a+\frac{36\cdots 67}{82\cdots 50}$ 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:  $C_{2}$, which has order $2$ (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:  $C_{2}\times C_{2}$, which has order $4$ (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:   $\frac{73\cdots 47}{18\cdots 00}a^{22}+\frac{73\cdots 47}{16\cdots 00}a^{21}-\frac{42\cdots 43}{18\cdots 00}a^{20}+\frac{27\cdots 67}{36\cdots 20}a^{19}-\frac{56\cdots 61}{36\cdots 20}a^{18}-\frac{20\cdots 61}{18\cdots 00}a^{17}+\frac{23\cdots 54}{45\cdots 75}a^{16}-\frac{94\cdots 47}{90\cdots 50}a^{15}-\frac{44\cdots 47}{36\cdots 20}a^{14}+\frac{34\cdots 17}{36\cdots 20}a^{13}-\frac{26\cdots 13}{18\cdots 00}a^{12}-\frac{61\cdots 07}{18\cdots 00}a^{11}+\frac{31\cdots 93}{18\cdots 00}a^{10}-\frac{12\cdots 47}{36\cdots 20}a^{9}+\frac{47\cdots 71}{18\cdots 11}a^{8}+\frac{17\cdots 13}{90\cdots 50}a^{7}-\frac{97\cdots 51}{18\cdots 00}a^{6}+\frac{58\cdots 49}{18\cdots 00}a^{5}-\frac{83\cdots 17}{72\cdots 44}a^{4}+\frac{70\cdots 89}{36\cdots 20}a^{3}-\frac{79\cdots 03}{45\cdots 75}a^{2}+\frac{53\cdots 81}{90\cdots 50}a-\frac{52\cdots 79}{90\cdots 50}$, $\frac{20\cdots 67}{18\cdots 00}a^{22}-\frac{20\cdots 67}{16\cdots 00}a^{21}+\frac{59\cdots 99}{90\cdots 50}a^{20}-\frac{78\cdots 37}{36\cdots 20}a^{19}+\frac{10\cdots 93}{18\cdots 10}a^{18}+\frac{57\cdots 71}{18\cdots 00}a^{17}-\frac{25\cdots 01}{18\cdots 00}a^{16}+\frac{27\cdots 17}{90\cdots 50}a^{15}+\frac{12\cdots 17}{36\cdots 20}a^{14}-\frac{94\cdots 97}{36\cdots 20}a^{13}+\frac{19\cdots 42}{45\cdots 75}a^{12}+\frac{15\cdots 77}{18\cdots 00}a^{11}-\frac{21\cdots 12}{45\cdots 75}a^{10}+\frac{34\cdots 67}{36\cdots 20}a^{9}-\frac{61\cdots 29}{72\cdots 44}a^{8}-\frac{30\cdots 43}{90\cdots 50}a^{7}+\frac{25\cdots 61}{18\cdots 00}a^{6}-\frac{20\cdots 89}{18\cdots 00}a^{5}+\frac{10\cdots 76}{18\cdots 11}a^{4}-\frac{20\cdots 89}{36\cdots 20}a^{3}+\frac{99\cdots 07}{18\cdots 00}a^{2}-\frac{11\cdots 83}{45\cdots 75}a+\frac{62\cdots 69}{90\cdots 50}$, $\frac{17\cdots 29}{18\cdots 00}a^{22}+\frac{17\cdots 29}{16\cdots 00}a^{21}-\frac{50\cdots 23}{90\cdots 50}a^{20}+\frac{65\cdots 59}{36\cdots 20}a^{19}-\frac{86\cdots 67}{36\cdots 20}a^{18}-\frac{12\cdots 63}{45\cdots 75}a^{17}+\frac{22\cdots 87}{18\cdots 00}a^{16}-\frac{22\cdots 09}{90\cdots 50}a^{15}-\frac{27\cdots 91}{90\cdots 55}a^{14}+\frac{40\cdots 77}{18\cdots 10}a^{13}-\frac{62\cdots 91}{18\cdots 00}a^{12}-\frac{78\cdots 37}{90\cdots 50}a^{11}+\frac{19\cdots 49}{45\cdots 75}a^{10}-\frac{28\cdots 99}{36\cdots 20}a^{9}+\frac{19\cdots 43}{36\cdots 22}a^{8}+\frac{11\cdots 57}{18\cdots 00}a^{7}-\frac{24\cdots 57}{18\cdots 00}a^{6}+\frac{92\cdots 53}{18\cdots 00}a^{5}+\frac{33\cdots 01}{18\cdots 11}a^{4}+\frac{33\cdots 03}{36\cdots 20}a^{3}-\frac{61\cdots 67}{90\cdots 50}a^{2}+\frac{95\cdots 09}{18\cdots 00}a-\frac{11\cdots 13}{90\cdots 50}$, $\frac{20\cdots 73}{36\cdots 20}a^{22}+\frac{20\cdots 73}{32\cdots 20}a^{21}-\frac{11\cdots 73}{36\cdots 22}a^{20}+\frac{79\cdots 39}{72\cdots 44}a^{19}-\frac{21\cdots 11}{72\cdots 44}a^{18}-\frac{14\cdots 91}{90\cdots 55}a^{17}+\frac{26\cdots 09}{36\cdots 20}a^{16}-\frac{54\cdots 37}{36\cdots 22}a^{15}-\frac{62\cdots 25}{36\cdots 22}a^{14}+\frac{23\cdots 23}{18\cdots 11}a^{13}-\frac{77\cdots 17}{36\cdots 20}a^{12}-\frac{40\cdots 32}{90\cdots 55}a^{11}+\frac{87\cdots 05}{36\cdots 22}a^{10}-\frac{35\cdots 33}{72\cdots 44}a^{9}+\frac{15\cdots 75}{36\cdots 22}a^{8}+\frac{67\cdots 39}{36\cdots 20}a^{7}-\frac{26\cdots 39}{36\cdots 20}a^{6}+\frac{40\cdots 55}{72\cdots 44}a^{5}-\frac{10\cdots 85}{36\cdots 22}a^{4}+\frac{20\cdots 61}{72\cdots 44}a^{3}-\frac{45\cdots 99}{18\cdots 10}a^{2}+\frac{39\cdots 53}{36\cdots 20}a-\frac{32\cdots 58}{18\cdots 11}$, $\frac{97\cdots 41}{83\cdots 40}a^{22}+\frac{97\cdots 41}{76\cdots 40}a^{21}-\frac{14\cdots 66}{20\cdots 85}a^{20}+\frac{37\cdots 71}{16\cdots 08}a^{19}-\frac{92\cdots 77}{16\cdots 08}a^{18}-\frac{69\cdots 82}{20\cdots 85}a^{17}+\frac{30\cdots 37}{20\cdots 85}a^{16}-\frac{12\cdots 21}{41\cdots 70}a^{15}-\frac{14\cdots 21}{41\cdots 77}a^{14}+\frac{22\cdots 69}{83\cdots 54}a^{13}-\frac{18\cdots 47}{41\cdots 70}a^{12}-\frac{19\cdots 69}{20\cdots 85}a^{11}+\frac{41\cdots 59}{83\cdots 40}a^{10}-\frac{16\cdots 73}{16\cdots 08}a^{9}+\frac{69\cdots 49}{83\cdots 54}a^{8}+\frac{35\cdots 83}{83\cdots 40}a^{7}-\frac{62\cdots 39}{41\cdots 70}a^{6}+\frac{89\cdots 47}{83\cdots 40}a^{5}-\frac{41\cdots 09}{83\cdots 54}a^{4}+\frac{94\cdots 15}{16\cdots 08}a^{3}-\frac{22\cdots 03}{41\cdots 70}a^{2}+\frac{20\cdots 91}{83\cdots 40}a-\frac{34\cdots 59}{83\cdots 40}$, $\frac{33\cdots 21}{45\cdots 75}a^{22}+\frac{33\cdots 21}{41\cdots 25}a^{21}-\frac{19\cdots 74}{45\cdots 75}a^{20}+\frac{12\cdots 31}{90\cdots 55}a^{19}-\frac{30\cdots 88}{90\cdots 55}a^{18}-\frac{96\cdots 23}{45\cdots 75}a^{17}+\frac{17\cdots 77}{18\cdots 00}a^{16}-\frac{89\cdots 92}{45\cdots 75}a^{15}-\frac{20\cdots 11}{90\cdots 55}a^{14}+\frac{15\cdots 46}{90\cdots 55}a^{13}-\frac{50\cdots 11}{18\cdots 00}a^{12}-\frac{54\cdots 77}{90\cdots 50}a^{11}+\frac{58\cdots 71}{18\cdots 00}a^{10}-\frac{11\cdots 87}{18\cdots 10}a^{9}+\frac{95\cdots 82}{18\cdots 11}a^{8}+\frac{34\cdots 11}{90\cdots 50}a^{7}-\frac{22\cdots 97}{18\cdots 00}a^{6}+\frac{48\cdots 32}{45\cdots 75}a^{5}-\frac{48\cdots 47}{18\cdots 11}a^{4}-\frac{36\cdots 01}{18\cdots 10}a^{3}+\frac{18\cdots 43}{90\cdots 50}a^{2}-\frac{67\cdots 93}{90\cdots 50}a+\frac{21\cdots 99}{18\cdots 00}$, $\frac{71\cdots 57}{90\cdots 50}a^{22}-\frac{71\cdots 57}{82\cdots 50}a^{21}+\frac{20\cdots 89}{45\cdots 75}a^{20}-\frac{26\cdots 67}{18\cdots 10}a^{19}+\frac{22\cdots 68}{90\cdots 55}a^{18}+\frac{20\cdots 91}{90\cdots 50}a^{17}-\frac{45\cdots 23}{45\cdots 75}a^{16}+\frac{91\cdots 87}{45\cdots 75}a^{15}+\frac{22\cdots 76}{90\cdots 55}a^{14}-\frac{16\cdots 51}{90\cdots 55}a^{13}+\frac{12\cdots 39}{45\cdots 75}a^{12}+\frac{30\cdots 71}{45\cdots 75}a^{11}-\frac{30\cdots 53}{90\cdots 50}a^{10}+\frac{11\cdots 17}{18\cdots 10}a^{9}-\frac{17\cdots 69}{36\cdots 22}a^{8}-\frac{37\cdots 31}{90\cdots 50}a^{7}+\frac{94\cdots 31}{90\cdots 50}a^{6}-\frac{25\cdots 77}{45\cdots 75}a^{5}+\frac{29\cdots 66}{18\cdots 11}a^{4}-\frac{68\cdots 59}{18\cdots 10}a^{3}+\frac{14\cdots 36}{45\cdots 75}a^{2}-\frac{38\cdots 86}{45\cdots 75}a+\frac{97\cdots 34}{45\cdots 75}$, $\frac{61\cdots 81}{18\cdots 00}a^{23}+\frac{69\cdots 09}{18\cdots 00}a^{22}-\frac{94\cdots 48}{45\cdots 75}a^{21}+\frac{32\cdots 43}{45\cdots 75}a^{20}-\frac{15\cdots 41}{36\cdots 20}a^{19}-\frac{42\cdots 52}{45\cdots 75}a^{18}+\frac{21\cdots 63}{45\cdots 75}a^{17}-\frac{19\cdots 31}{18\cdots 00}a^{16}-\frac{89\cdots 47}{90\cdots 50}a^{15}+\frac{29\cdots 39}{36\cdots 20}a^{14}-\frac{71\cdots 66}{45\cdots 75}a^{13}-\frac{19\cdots 57}{90\cdots 50}a^{12}+\frac{27\cdots 27}{18\cdots 00}a^{11}-\frac{30\cdots 61}{90\cdots 50}a^{10}+\frac{58\cdots 79}{16\cdots 10}a^{9}+\frac{36\cdots 23}{18\cdots 00}a^{8}-\frac{21\cdots 23}{45\cdots 75}a^{7}+\frac{86\cdots 11}{18\cdots 00}a^{6}-\frac{11\cdots 74}{41\cdots 25}a^{5}+\frac{21\cdots 88}{90\cdots 55}a^{4}-\frac{20\cdots 03}{90\cdots 50}a^{3}+\frac{24\cdots 79}{18\cdots 00}a^{2}-\frac{84\cdots 67}{18\cdots 00}a+\frac{63\cdots 51}{90\cdots 50}$, $\frac{90\cdots 21}{90\cdots 50}a^{23}+\frac{40\cdots 63}{36\cdots 20}a^{22}-\frac{27\cdots 09}{45\cdots 75}a^{21}+\frac{86\cdots 29}{41\cdots 00}a^{20}-\frac{32\cdots 19}{36\cdots 20}a^{19}-\frac{12\cdots 19}{45\cdots 75}a^{18}+\frac{47\cdots 93}{36\cdots 20}a^{17}-\frac{52\cdots 33}{18\cdots 00}a^{16}-\frac{54\cdots 39}{18\cdots 00}a^{15}+\frac{43\cdots 28}{18\cdots 11}a^{14}-\frac{38\cdots 79}{90\cdots 50}a^{13}-\frac{25\cdots 63}{36\cdots 20}a^{12}+\frac{79\cdots 21}{18\cdots 00}a^{11}-\frac{84\cdots 11}{90\cdots 50}a^{10}+\frac{33\cdots 13}{36\cdots 20}a^{9}+\frac{29\cdots 11}{18\cdots 00}a^{8}-\frac{47\cdots 27}{36\cdots 20}a^{7}+\frac{55\cdots 07}{45\cdots 75}a^{6}-\frac{32\cdots 04}{45\cdots 75}a^{5}+\frac{22\cdots 59}{36\cdots 20}a^{4}-\frac{10\cdots 87}{18\cdots 00}a^{3}+\frac{12\cdots 59}{36\cdots 20}a^{2}-\frac{19\cdots 31}{18\cdots 00}a+\frac{28\cdots 97}{18\cdots 00}$, $\frac{26\cdots 63}{16\cdots 00}a^{23}-\frac{16\cdots 21}{90\cdots 50}a^{22}+\frac{15\cdots 21}{16\cdots 00}a^{21}-\frac{29\cdots 13}{90\cdots 50}a^{20}+\frac{14\cdots 02}{90\cdots 55}a^{19}+\frac{80\cdots 49}{18\cdots 00}a^{18}-\frac{19\cdots 13}{90\cdots 50}a^{17}+\frac{85\cdots 33}{18\cdots 00}a^{16}+\frac{21\cdots 63}{45\cdots 75}a^{15}-\frac{13\cdots 07}{36\cdots 20}a^{14}+\frac{31\cdots 48}{45\cdots 75}a^{13}+\frac{49\cdots 33}{45\cdots 75}a^{12}-\frac{29\cdots 27}{41\cdots 00}a^{11}+\frac{27\cdots 51}{18\cdots 00}a^{10}-\frac{54\cdots 29}{36\cdots 20}a^{9}-\frac{39\cdots 69}{18\cdots 00}a^{8}+\frac{38\cdots 21}{18\cdots 00}a^{7}-\frac{36\cdots 73}{18\cdots 00}a^{6}+\frac{21\cdots 73}{18\cdots 00}a^{5}-\frac{18\cdots 63}{18\cdots 10}a^{4}+\frac{17\cdots 93}{18\cdots 00}a^{3}-\frac{10\cdots 77}{18\cdots 00}a^{2}+\frac{83\cdots 14}{45\cdots 75}a-\frac{48\cdots 41}{18\cdots 00}$, $\frac{29\cdots 29}{18\cdots 00}a^{23}+\frac{37\cdots 53}{18\cdots 00}a^{22}-\frac{88\cdots 87}{72\cdots 44}a^{21}+\frac{81\cdots 31}{18\cdots 00}a^{20}-\frac{18\cdots 61}{36\cdots 20}a^{19}-\frac{21\cdots 98}{45\cdots 75}a^{18}+\frac{12\cdots 01}{45\cdots 75}a^{17}-\frac{26\cdots 07}{36\cdots 20}a^{16}-\frac{80\cdots 07}{18\cdots 00}a^{15}+\frac{41\cdots 22}{90\cdots 55}a^{14}-\frac{19\cdots 51}{16\cdots 00}a^{13}-\frac{99\cdots 13}{18\cdots 00}a^{12}+\frac{33\cdots 01}{36\cdots 20}a^{11}-\frac{21\cdots 53}{90\cdots 50}a^{10}+\frac{27\cdots 41}{90\cdots 55}a^{9}-\frac{13\cdots 43}{18\cdots 00}a^{8}-\frac{57\cdots 29}{18\cdots 00}a^{7}+\frac{37\cdots 62}{90\cdots 55}a^{6}-\frac{42\cdots 73}{18\cdots 00}a^{5}+\frac{64\cdots 73}{36\cdots 20}a^{4}-\frac{82\cdots 56}{45\cdots 75}a^{3}+\frac{22\cdots 23}{18\cdots 00}a^{2}-\frac{41\cdots 91}{90\cdots 55}a+\frac{13\cdots 41}{18\cdots 00}$, $\frac{93\cdots 93}{18\cdots 00}a^{23}+\frac{10\cdots 67}{16\cdots 00}a^{22}-\frac{35\cdots 13}{90\cdots 50}a^{21}+\frac{67\cdots 99}{45\cdots 75}a^{20}-\frac{33\cdots 19}{18\cdots 10}a^{19}-\frac{13\cdots 37}{90\cdots 50}a^{18}+\frac{16\cdots 61}{18\cdots 00}a^{17}-\frac{44\cdots 43}{18\cdots 00}a^{16}-\frac{24\cdots 17}{18\cdots 00}a^{15}+\frac{53\cdots 97}{36\cdots 20}a^{14}-\frac{35\cdots 71}{90\cdots 50}a^{13}-\frac{11\cdots 01}{90\cdots 50}a^{12}+\frac{13\cdots 64}{45\cdots 75}a^{11}-\frac{35\cdots 99}{45\cdots 75}a^{10}+\frac{37\cdots 09}{36\cdots 20}a^{9}-\frac{56\cdots 31}{18\cdots 00}a^{8}-\frac{19\cdots 81}{18\cdots 00}a^{7}+\frac{44\cdots 87}{30\cdots 00}a^{6}-\frac{88\cdots 89}{10\cdots 25}a^{5}+\frac{11\cdots 53}{18\cdots 10}a^{4}-\frac{11\cdots 43}{18\cdots 00}a^{3}+\frac{78\cdots 47}{18\cdots 00}a^{2}-\frac{75\cdots 94}{45\cdots 75}a+\frac{11\cdots 19}{41\cdots 25}$, $\frac{22\cdots 28}{90\cdots 55}a^{23}+\frac{12\cdots 18}{45\cdots 75}a^{22}-\frac{28\cdots 27}{18\cdots 00}a^{21}+\frac{51\cdots 69}{90\cdots 50}a^{20}-\frac{36\cdots 83}{82\cdots 05}a^{19}-\frac{24\cdots 21}{36\cdots 22}a^{18}+\frac{64\cdots 61}{18\cdots 00}a^{17}-\frac{39\cdots 74}{45\cdots 75}a^{16}-\frac{12\cdots 71}{18\cdots 00}a^{15}+\frac{10\cdots 31}{16\cdots 10}a^{14}-\frac{23\cdots 06}{18\cdots 11}a^{13}-\frac{58\cdots 28}{45\cdots 75}a^{12}+\frac{21\cdots 17}{18\cdots 00}a^{11}-\frac{58\cdots 83}{20\cdots 50}a^{10}+\frac{57\cdots 81}{18\cdots 10}a^{9}-\frac{58\cdots 97}{18\cdots 10}a^{8}-\frac{17\cdots 69}{45\cdots 75}a^{7}+\frac{20\cdots 89}{45\cdots 75}a^{6}-\frac{46\cdots 59}{18\cdots 00}a^{5}+\frac{33\cdots 95}{18\cdots 11}a^{4}-\frac{35\cdots 73}{18\cdots 10}a^{3}+\frac{58\cdots 53}{45\cdots 75}a^{2}-\frac{18\cdots 93}{45\cdots 75}a+\frac{43\cdots 89}{90\cdots 50}$ 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:  \( 1585199492413569000 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 1585199492413569000 \cdot 2}{2\cdot\sqrt{54372494083060640194576311797602218575775623321533203125}}\cr\approx \mathstrut & 0.329847322806998 \end{aligned}\] (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 - 12*x^23 + 69*x^22 - 253*x^21 + 261*x^20 + 2703*x^19 - 15406*x^18 + 39987*x^17 + 272986*x^16 - 2610282*x^15 + 6157629*x^14 + 3326657*x^13 - 48975994*x^12 + 129522333*x^11 - 171855021*x^10 + 68435337*x^9 + 134996941*x^8 - 232660062*x^7 + 184844654*x^6 - 129342673*x^5 + 113722606*x^4 - 86398607*x^3 + 43132904*x^2 - 12596758*x + 1680671) 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 - 12*x^23 + 69*x^22 - 253*x^21 + 261*x^20 + 2703*x^19 - 15406*x^18 + 39987*x^17 + 272986*x^16 - 2610282*x^15 + 6157629*x^14 + 3326657*x^13 - 48975994*x^12 + 129522333*x^11 - 171855021*x^10 + 68435337*x^9 + 134996941*x^8 - 232660062*x^7 + 184844654*x^6 - 129342673*x^5 + 113722606*x^4 - 86398607*x^3 + 43132904*x^2 - 12596758*x + 1680671, 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 - 12*x^23 + 69*x^22 - 253*x^21 + 261*x^20 + 2703*x^19 - 15406*x^18 + 39987*x^17 + 272986*x^16 - 2610282*x^15 + 6157629*x^14 + 3326657*x^13 - 48975994*x^12 + 129522333*x^11 - 171855021*x^10 + 68435337*x^9 + 134996941*x^8 - 232660062*x^7 + 184844654*x^6 - 129342673*x^5 + 113722606*x^4 - 86398607*x^3 + 43132904*x^2 - 12596758*x + 1680671); 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 - 12*x^23 + 69*x^22 - 253*x^21 + 261*x^20 + 2703*x^19 - 15406*x^18 + 39987*x^17 + 272986*x^16 - 2610282*x^15 + 6157629*x^14 + 3326657*x^13 - 48975994*x^12 + 129522333*x^11 - 171855021*x^10 + 68435337*x^9 + 134996941*x^8 - 232660062*x^7 + 184844654*x^6 - 129342673*x^5 + 113722606*x^4 - 86398607*x^3 + 43132904*x^2 - 12596758*x + 1680671); 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

6.2.267093828125.2, 12.4.356695565112335205078125.1

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: 24.4.2174899763322425607783052471904088743031024932861328125.3, 24.4.2174899763322425607783052471904088743031024932861328125.5
Arithmetically equivalent sibling: 24.4.54372494083060640194576311797602218575775623321533203125.5
Minimal sibling: 24.4.2174899763322425607783052471904088743031024932861328125.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{3}$ $24$ R $24$ ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{3}$ ${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{10}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{3}$ ${\href{/padicField/29.12.0.1}{12} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{3}$ ${\href{/padicField/41.5.0.1}{5} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ R ${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{10}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 5.2.2.2a1.1$x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
5.1.20.31a2.17$x^{20} + 10 x^{12} + 15$$20$$1$$31$20T20$not computed$
\(43\) Copy content Toggle raw display 43.2.2.2a1.1$x^{4} + 84 x^{3} + 1770 x^{2} + 295 x + 9$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
43.1.10.9a1.2$x^{10} + 129$$10$$1$$9$$F_{5}\times C_2$$$[\ ]_{10}^{4}$$
43.1.10.9a1.2$x^{10} + 129$$10$$1$$9$$F_{5}\times C_2$$$[\ ]_{10}^{4}$$

Spectrum of ring of integers

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