Properties

Label 24.0.221...848.1
Degree $24$
Signature $(0, 12)$
Discriminant $2.220\times 10^{37}$
Root discriminant \(35.98\)
Ramified primes $2,257$
Class number $4$ (GRH)
Class group [2, 2] (GRH)
Galois group $C_3:D_8$ (as 24T37)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^23 + 8*x^22 - 10*x^21 + 70*x^20 - 182*x^19 + 269*x^18 - 372*x^17 + 1647*x^16 - 1870*x^15 + 851*x^14 - 1270*x^13 + 4539*x^12 + 10226*x^11 - 6485*x^10 - 6130*x^9 - 7385*x^8 + 932*x^7 + 26931*x^6 + 14618*x^5 - 4894*x^4 + 334*x^3 + 2048*x^2 - 312*x + 41)
 
Copy content gp:K = bnfinit(y^24 - 4*y^23 + 8*y^22 - 10*y^21 + 70*y^20 - 182*y^19 + 269*y^18 - 372*y^17 + 1647*y^16 - 1870*y^15 + 851*y^14 - 1270*y^13 + 4539*y^12 + 10226*y^11 - 6485*y^10 - 6130*y^9 - 7385*y^8 + 932*y^7 + 26931*y^6 + 14618*y^5 - 4894*y^4 + 334*y^3 + 2048*y^2 - 312*y + 41, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 4*x^23 + 8*x^22 - 10*x^21 + 70*x^20 - 182*x^19 + 269*x^18 - 372*x^17 + 1647*x^16 - 1870*x^15 + 851*x^14 - 1270*x^13 + 4539*x^12 + 10226*x^11 - 6485*x^10 - 6130*x^9 - 7385*x^8 + 932*x^7 + 26931*x^6 + 14618*x^5 - 4894*x^4 + 334*x^3 + 2048*x^2 - 312*x + 41);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 - 4*x^23 + 8*x^22 - 10*x^21 + 70*x^20 - 182*x^19 + 269*x^18 - 372*x^17 + 1647*x^16 - 1870*x^15 + 851*x^14 - 1270*x^13 + 4539*x^12 + 10226*x^11 - 6485*x^10 - 6130*x^9 - 7385*x^8 + 932*x^7 + 26931*x^6 + 14618*x^5 - 4894*x^4 + 334*x^3 + 2048*x^2 - 312*x + 41)
 

\( x^{24} - 4 x^{23} + 8 x^{22} - 10 x^{21} + 70 x^{20} - 182 x^{19} + 269 x^{18} - 372 x^{17} + 1647 x^{16} + \cdots + 41 \) 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:  $(0, 12)$
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:   \(22199551505513271594529924008114126848\) \(\medspace = 2^{36}\cdot 257^{11}\) 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:  \(35.98\)
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:   \(2\), \(257\) 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:  \(\Q(\sqrt{257}) \)
$\Aut(K/\Q)$:   $C_2$
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 a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{7}a^{20}-\frac{1}{7}a^{19}+\frac{1}{7}a^{18}-\frac{3}{7}a^{17}-\frac{2}{7}a^{16}+\frac{2}{7}a^{15}+\frac{1}{7}a^{12}+\frac{3}{7}a^{11}+\frac{2}{7}a^{10}-\frac{2}{7}a^{9}-\frac{2}{7}a^{7}-\frac{1}{7}a^{6}-\frac{1}{7}a^{5}+\frac{1}{7}a^{4}+\frac{1}{7}a^{2}+\frac{1}{7}a+\frac{2}{7}$, $\frac{1}{287}a^{21}-\frac{2}{287}a^{20}+\frac{37}{287}a^{19}-\frac{95}{287}a^{18}+\frac{57}{287}a^{17}+\frac{39}{287}a^{16}+\frac{75}{287}a^{15}+\frac{4}{41}a^{14}+\frac{141}{287}a^{13}-\frac{54}{287}a^{12}-\frac{78}{287}a^{11}+\frac{38}{287}a^{10}+\frac{93}{287}a^{9}+\frac{19}{287}a^{8}+\frac{99}{287}a^{7}-\frac{2}{41}a^{6}+\frac{114}{287}a^{5}-\frac{22}{287}a^{4}-\frac{111}{287}a^{3}-\frac{1}{41}a^{2}-\frac{6}{287}a+\frac{2}{7}$, $\frac{1}{353297}a^{22}+\frac{461}{353297}a^{21}+\frac{13912}{353297}a^{20}+\frac{121914}{353297}a^{19}-\frac{7400}{50471}a^{18}-\frac{46960}{353297}a^{17}+\frac{86397}{353297}a^{16}-\frac{157496}{353297}a^{15}+\frac{20854}{353297}a^{14}-\frac{56459}{353297}a^{13}+\frac{32484}{353297}a^{12}+\frac{41332}{353297}a^{11}-\frac{172266}{353297}a^{10}+\frac{19503}{353297}a^{9}+\frac{44771}{353297}a^{8}-\frac{104893}{353297}a^{7}+\frac{120035}{353297}a^{6}-\frac{66796}{353297}a^{5}+\frac{152883}{353297}a^{4}+\frac{68853}{353297}a^{3}-\frac{134242}{353297}a^{2}-\frac{80596}{353297}a-\frac{109}{8617}$, $\frac{1}{24\cdots 09}a^{23}+\frac{29\cdots 05}{24\cdots 09}a^{22}-\frac{21\cdots 11}{34\cdots 87}a^{21}+\frac{56\cdots 04}{24\cdots 09}a^{20}+\frac{11\cdots 69}{24\cdots 09}a^{19}-\frac{11\cdots 45}{24\cdots 09}a^{18}+\frac{57\cdots 56}{24\cdots 09}a^{17}-\frac{73\cdots 32}{24\cdots 09}a^{16}-\frac{14\cdots 14}{24\cdots 09}a^{15}+\frac{11\cdots 47}{24\cdots 09}a^{14}-\frac{86\cdots 12}{24\cdots 09}a^{13}-\frac{12\cdots 34}{24\cdots 09}a^{12}-\frac{11\cdots 74}{24\cdots 09}a^{11}-\frac{11\cdots 95}{24\cdots 09}a^{10}+\frac{11\cdots 89}{24\cdots 09}a^{9}+\frac{11\cdots 59}{24\cdots 09}a^{8}-\frac{67\cdots 13}{24\cdots 09}a^{7}+\frac{49\cdots 17}{34\cdots 87}a^{6}+\frac{29\cdots 24}{24\cdots 09}a^{5}-\frac{34\cdots 80}{24\cdots 09}a^{4}+\frac{68\cdots 19}{34\cdots 87}a^{3}-\frac{14\cdots 76}{24\cdots 09}a^{2}-\frac{11\cdots 32}{24\cdots 09}a+\frac{29\cdots 65}{59\cdots 49}$ 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:  $C_{2}\times C_{2}$, 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:  $C_{2}\times C_{2}$, 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);
 
Relative class number:   data not computed (assuming GRH)

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:  $11$
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:   $\frac{29\cdots 88}{20\cdots 27}a^{23}-\frac{96\cdots 54}{20\cdots 27}a^{22}+\frac{15\cdots 19}{20\cdots 27}a^{21}-\frac{21\cdots 53}{28\cdots 61}a^{20}+\frac{19\cdots 04}{20\cdots 27}a^{19}-\frac{56\cdots 91}{28\cdots 61}a^{18}+\frac{44\cdots 43}{20\cdots 27}a^{17}-\frac{64\cdots 90}{20\cdots 27}a^{16}+\frac{42\cdots 35}{20\cdots 27}a^{15}-\frac{31\cdots 25}{28\cdots 61}a^{14}-\frac{62\cdots 39}{28\cdots 61}a^{13}-\frac{32\cdots 93}{20\cdots 27}a^{12}+\frac{11\cdots 89}{20\cdots 27}a^{11}+\frac{39\cdots 80}{20\cdots 27}a^{10}+\frac{84\cdots 32}{28\cdots 61}a^{9}-\frac{25\cdots 08}{20\cdots 27}a^{8}-\frac{40\cdots 24}{20\cdots 27}a^{7}-\frac{18\cdots 78}{20\cdots 27}a^{6}+\frac{77\cdots 52}{20\cdots 27}a^{5}+\frac{10\cdots 47}{20\cdots 27}a^{4}+\frac{35\cdots 52}{20\cdots 27}a^{3}+\frac{56\cdots 16}{28\cdots 61}a^{2}+\frac{17\cdots 46}{20\cdots 27}a+\frac{94\cdots 64}{20\cdots 27}$, $\frac{17\cdots 54}{30\cdots 39}a^{23}+\frac{28\cdots 10}{30\cdots 39}a^{22}+\frac{20\cdots 20}{30\cdots 39}a^{21}-\frac{13\cdots 49}{30\cdots 39}a^{20}-\frac{21\cdots 08}{73\cdots 79}a^{19}+\frac{34\cdots 55}{30\cdots 39}a^{18}+\frac{35\cdots 58}{43\cdots 77}a^{17}-\frac{35\cdots 59}{30\cdots 39}a^{16}-\frac{15\cdots 46}{30\cdots 39}a^{15}-\frac{33\cdots 26}{30\cdots 39}a^{14}+\frac{53\cdots 54}{30\cdots 39}a^{13}-\frac{18\cdots 84}{30\cdots 39}a^{12}-\frac{46\cdots 10}{43\cdots 77}a^{11}-\frac{36\cdots 50}{30\cdots 39}a^{10}-\frac{34\cdots 90}{30\cdots 39}a^{9}+\frac{45\cdots 15}{43\cdots 77}a^{8}+\frac{10\cdots 14}{73\cdots 79}a^{7}+\frac{46\cdots 73}{43\cdots 77}a^{6}-\frac{47\cdots 44}{30\cdots 39}a^{5}-\frac{13\cdots 38}{30\cdots 39}a^{4}-\frac{67\cdots 48}{30\cdots 39}a^{3}+\frac{11\cdots 00}{30\cdots 39}a^{2}-\frac{19\cdots 74}{30\cdots 39}a-\frac{30\cdots 39}{10\cdots 97}$, $\frac{74\cdots 96}{24\cdots 09}a^{23}+\frac{37\cdots 85}{24\cdots 09}a^{22}-\frac{94\cdots 71}{24\cdots 09}a^{21}+\frac{15\cdots 51}{24\cdots 09}a^{20}-\frac{65\cdots 39}{24\cdots 09}a^{19}+\frac{19\cdots 37}{24\cdots 09}a^{18}-\frac{37\cdots 04}{24\cdots 09}a^{17}+\frac{60\cdots 44}{24\cdots 09}a^{16}-\frac{17\cdots 61}{24\cdots 09}a^{15}+\frac{29\cdots 63}{24\cdots 09}a^{14}-\frac{31\cdots 76}{24\cdots 09}a^{13}+\frac{33\cdots 82}{24\cdots 09}a^{12}-\frac{60\cdots 24}{24\cdots 09}a^{11}-\frac{23\cdots 38}{24\cdots 09}a^{10}+\frac{87\cdots 38}{24\cdots 09}a^{9}-\frac{28\cdots 70}{24\cdots 09}a^{8}+\frac{68\cdots 76}{24\cdots 09}a^{7}-\frac{77\cdots 81}{24\cdots 09}a^{6}-\frac{14\cdots 19}{24\cdots 09}a^{5}+\frac{49\cdots 55}{24\cdots 09}a^{4}+\frac{34\cdots 42}{24\cdots 09}a^{3}-\frac{24\cdots 82}{24\cdots 09}a^{2}+\frac{51\cdots 58}{34\cdots 87}a-\frac{46\cdots 33}{59\cdots 49}$, $\frac{68\cdots 64}{34\cdots 87}a^{23}-\frac{22\cdots 92}{24\cdots 09}a^{22}+\frac{53\cdots 31}{24\cdots 09}a^{21}-\frac{84\cdots 95}{24\cdots 09}a^{20}+\frac{39\cdots 63}{24\cdots 09}a^{19}-\frac{16\cdots 88}{34\cdots 87}a^{18}+\frac{20\cdots 73}{24\cdots 09}a^{17}-\frac{31\cdots 29}{24\cdots 09}a^{16}+\frac{10\cdots 52}{24\cdots 09}a^{15}-\frac{15\cdots 49}{24\cdots 09}a^{14}+\frac{14\cdots 17}{24\cdots 09}a^{13}-\frac{16\cdots 52}{24\cdots 09}a^{12}+\frac{32\cdots 58}{24\cdots 09}a^{11}+\frac{26\cdots 07}{24\cdots 09}a^{10}-\frac{49\cdots 20}{24\cdots 09}a^{9}+\frac{57\cdots 76}{24\cdots 09}a^{8}-\frac{39\cdots 47}{24\cdots 09}a^{7}+\frac{46\cdots 87}{34\cdots 87}a^{6}+\frac{10\cdots 84}{24\cdots 09}a^{5}-\frac{44\cdots 75}{24\cdots 09}a^{4}-\frac{22\cdots 66}{24\cdots 09}a^{3}+\frac{17\cdots 27}{24\cdots 09}a^{2}-\frac{25\cdots 06}{24\cdots 09}a+\frac{33\cdots 04}{59\cdots 49}$, $\frac{13\cdots 27}{34\cdots 87}a^{23}+\frac{22\cdots 01}{24\cdots 09}a^{22}-\frac{18\cdots 66}{24\cdots 09}a^{21}-\frac{15\cdots 09}{24\cdots 09}a^{20}-\frac{52\cdots 28}{24\cdots 09}a^{19}+\frac{69\cdots 84}{24\cdots 09}a^{18}+\frac{27\cdots 43}{34\cdots 87}a^{17}-\frac{92\cdots 65}{24\cdots 09}a^{16}-\frac{10\cdots 85}{24\cdots 09}a^{15}-\frac{60\cdots 94}{24\cdots 09}a^{14}+\frac{16\cdots 61}{24\cdots 09}a^{13}+\frac{25\cdots 64}{24\cdots 09}a^{12}-\frac{84\cdots 72}{85\cdots 07}a^{11}-\frac{16\cdots 75}{24\cdots 09}a^{10}-\frac{97\cdots 86}{24\cdots 09}a^{9}+\frac{12\cdots 63}{24\cdots 09}a^{8}+\frac{24\cdots 29}{34\cdots 87}a^{7}+\frac{12\cdots 47}{24\cdots 09}a^{6}-\frac{25\cdots 05}{24\cdots 09}a^{5}-\frac{53\cdots 03}{24\cdots 09}a^{4}-\frac{23\cdots 19}{24\cdots 09}a^{3}+\frac{27\cdots 86}{24\cdots 09}a^{2}-\frac{51\cdots 20}{24\cdots 09}a-\frac{45\cdots 73}{59\cdots 49}$, $\frac{48\cdots 18}{24\cdots 09}a^{23}+\frac{19\cdots 33}{34\cdots 87}a^{22}-\frac{16\cdots 33}{24\cdots 09}a^{21}+\frac{52\cdots 51}{24\cdots 09}a^{20}-\frac{28\cdots 21}{24\cdots 09}a^{19}+\frac{48\cdots 12}{24\cdots 09}a^{18}-\frac{30\cdots 77}{24\cdots 09}a^{17}+\frac{36\cdots 48}{24\cdots 09}a^{16}-\frac{59\cdots 45}{24\cdots 09}a^{15}-\frac{33\cdots 90}{24\cdots 09}a^{14}+\frac{57\cdots 25}{24\cdots 09}a^{13}+\frac{18\cdots 24}{24\cdots 09}a^{12}-\frac{13\cdots 22}{24\cdots 09}a^{11}-\frac{76\cdots 60}{24\cdots 09}a^{10}-\frac{30\cdots 87}{24\cdots 09}a^{9}+\frac{57\cdots 57}{24\cdots 09}a^{8}+\frac{75\cdots 48}{24\cdots 09}a^{7}+\frac{51\cdots 30}{24\cdots 09}a^{6}-\frac{13\cdots 39}{24\cdots 09}a^{5}-\frac{22\cdots 24}{24\cdots 09}a^{4}-\frac{12\cdots 83}{34\cdots 87}a^{3}+\frac{60\cdots 25}{24\cdots 09}a^{2}-\frac{13\cdots 00}{24\cdots 09}a-\frac{45\cdots 83}{85\cdots 07}$, $\frac{10\cdots 40}{24\cdots 09}a^{23}+\frac{55\cdots 47}{24\cdots 09}a^{22}-\frac{32\cdots 72}{34\cdots 87}a^{21}+\frac{48\cdots 00}{24\cdots 09}a^{20}-\frac{66\cdots 14}{24\cdots 09}a^{19}+\frac{58\cdots 27}{34\cdots 87}a^{18}-\frac{10\cdots 83}{24\cdots 09}a^{17}+\frac{17\cdots 93}{24\cdots 09}a^{16}-\frac{25\cdots 08}{24\cdots 09}a^{15}+\frac{99\cdots 84}{24\cdots 09}a^{14}-\frac{17\cdots 36}{34\cdots 87}a^{13}+\frac{86\cdots 51}{24\cdots 09}a^{12}-\frac{11\cdots 17}{24\cdots 09}a^{11}+\frac{30\cdots 42}{24\cdots 09}a^{10}+\frac{49\cdots 43}{24\cdots 09}a^{9}-\frac{53\cdots 02}{34\cdots 87}a^{8}-\frac{22\cdots 97}{24\cdots 09}a^{7}-\frac{48\cdots 44}{24\cdots 09}a^{6}+\frac{15\cdots 04}{24\cdots 09}a^{5}+\frac{13\cdots 76}{24\cdots 09}a^{4}+\frac{65\cdots 14}{24\cdots 09}a^{3}-\frac{13\cdots 77}{24\cdots 09}a^{2}+\frac{32\cdots 52}{34\cdots 87}a+\frac{19\cdots 88}{59\cdots 49}$, $\frac{16\cdots 50}{24\cdots 09}a^{23}-\frac{11\cdots 60}{24\cdots 09}a^{22}+\frac{34\cdots 24}{24\cdots 09}a^{21}-\frac{67\cdots 74}{24\cdots 09}a^{20}+\frac{19\cdots 70}{24\cdots 09}a^{19}-\frac{68\cdots 54}{24\cdots 09}a^{18}+\frac{15\cdots 42}{24\cdots 09}a^{17}-\frac{63\cdots 16}{59\cdots 49}a^{16}+\frac{58\cdots 89}{24\cdots 09}a^{15}-\frac{10\cdots 77}{19\cdots 39}a^{14}+\frac{23\cdots 35}{34\cdots 87}a^{13}-\frac{16\cdots 60}{24\cdots 09}a^{12}+\frac{24\cdots 31}{24\cdots 09}a^{11}-\frac{24\cdots 01}{34\cdots 87}a^{10}-\frac{37\cdots 97}{24\cdots 09}a^{9}+\frac{32\cdots 84}{24\cdots 09}a^{8}-\frac{16\cdots 34}{24\cdots 09}a^{7}+\frac{51\cdots 77}{24\cdots 09}a^{6}+\frac{14\cdots 19}{34\cdots 87}a^{5}-\frac{76\cdots 85}{24\cdots 09}a^{4}-\frac{20\cdots 74}{24\cdots 09}a^{3}+\frac{49\cdots 47}{24\cdots 09}a^{2}-\frac{89\cdots 32}{24\cdots 09}a+\frac{13\cdots 41}{59\cdots 49}$, $\frac{54\cdots 96}{24\cdots 09}a^{23}+\frac{14\cdots 62}{24\cdots 09}a^{22}-\frac{16\cdots 72}{24\cdots 09}a^{21}+\frac{18\cdots 06}{24\cdots 09}a^{20}-\frac{31\cdots 39}{24\cdots 09}a^{19}+\frac{50\cdots 69}{24\cdots 09}a^{18}-\frac{35\cdots 19}{34\cdots 87}a^{17}+\frac{29\cdots 32}{24\cdots 09}a^{16}-\frac{65\cdots 36}{24\cdots 09}a^{15}-\frac{11\cdots 29}{24\cdots 09}a^{14}+\frac{71\cdots 38}{24\cdots 09}a^{13}+\frac{22\cdots 60}{24\cdots 09}a^{12}-\frac{21\cdots 76}{34\cdots 87}a^{11}-\frac{21\cdots 14}{59\cdots 49}a^{10}-\frac{57\cdots 16}{34\cdots 87}a^{9}+\frac{67\cdots 67}{24\cdots 09}a^{8}+\frac{12\cdots 31}{34\cdots 87}a^{7}+\frac{60\cdots 10}{24\cdots 09}a^{6}-\frac{14\cdots 75}{24\cdots 09}a^{5}-\frac{27\cdots 27}{24\cdots 09}a^{4}-\frac{11\cdots 17}{24\cdots 09}a^{3}+\frac{98\cdots 81}{24\cdots 09}a^{2}-\frac{20\cdots 24}{24\cdots 09}a-\frac{10\cdots 38}{59\cdots 49}$, $\frac{50\cdots 30}{24\cdots 09}a^{23}-\frac{19\cdots 75}{24\cdots 09}a^{22}+\frac{57\cdots 51}{34\cdots 87}a^{21}-\frac{56\cdots 38}{24\cdots 09}a^{20}+\frac{36\cdots 50}{24\cdots 09}a^{19}-\frac{91\cdots 73}{24\cdots 09}a^{18}+\frac{14\cdots 78}{24\cdots 09}a^{17}-\frac{22\cdots 94}{24\cdots 09}a^{16}+\frac{89\cdots 79}{24\cdots 09}a^{15}-\frac{14\cdots 29}{34\cdots 87}a^{14}+\frac{77\cdots 87}{24\cdots 09}a^{13}-\frac{12\cdots 78}{24\cdots 09}a^{12}+\frac{27\cdots 50}{24\cdots 09}a^{11}+\frac{47\cdots 68}{24\cdots 09}a^{10}-\frac{14\cdots 45}{24\cdots 09}a^{9}-\frac{18\cdots 77}{24\cdots 09}a^{8}-\frac{61\cdots 05}{24\cdots 09}a^{7}+\frac{52\cdots 77}{24\cdots 09}a^{6}+\frac{11\cdots 44}{24\cdots 09}a^{5}+\frac{98\cdots 51}{24\cdots 09}a^{4}+\frac{32\cdots 58}{24\cdots 09}a^{3}+\frac{53\cdots 81}{24\cdots 09}a^{2}-\frac{54\cdots 88}{24\cdots 09}a+\frac{94\cdots 56}{85\cdots 07}$, $\frac{56\cdots 77}{24\cdots 09}a^{23}-\frac{32\cdots 65}{24\cdots 09}a^{22}+\frac{98\cdots 40}{24\cdots 09}a^{21}-\frac{20\cdots 00}{24\cdots 09}a^{20}+\frac{68\cdots 21}{24\cdots 09}a^{19}-\frac{20\cdots 50}{24\cdots 09}a^{18}+\frac{45\cdots 42}{24\cdots 09}a^{17}-\frac{86\cdots 99}{24\cdots 09}a^{16}+\frac{20\cdots 05}{24\cdots 09}a^{15}-\frac{38\cdots 53}{24\cdots 09}a^{14}+\frac{57\cdots 80}{24\cdots 09}a^{13}-\frac{74\cdots 59}{24\cdots 09}a^{12}+\frac{99\cdots 69}{24\cdots 09}a^{11}-\frac{64\cdots 73}{34\cdots 87}a^{10}-\frac{51\cdots 44}{59\cdots 49}a^{9}+\frac{87\cdots 85}{24\cdots 09}a^{8}-\frac{25\cdots 13}{34\cdots 87}a^{7}+\frac{16\cdots 46}{24\cdots 09}a^{6}+\frac{24\cdots 64}{34\cdots 87}a^{5}-\frac{62\cdots 72}{24\cdots 09}a^{4}+\frac{11\cdots 59}{24\cdots 09}a^{3}-\frac{47\cdots 87}{34\cdots 87}a^{2}+\frac{45\cdots 02}{24\cdots 09}a+\frac{62\cdots 74}{59\cdots 49}$ 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:  \( 145811959.84692615 \) (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)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 145811959.84692615 \cdot 4}{2\cdot\sqrt{22199551505513271594529924008114126848}}\cr\approx \mathstrut & 0.234320056291893 \end{aligned}\] (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 - 4*x^23 + 8*x^22 - 10*x^21 + 70*x^20 - 182*x^19 + 269*x^18 - 372*x^17 + 1647*x^16 - 1870*x^15 + 851*x^14 - 1270*x^13 + 4539*x^12 + 10226*x^11 - 6485*x^10 - 6130*x^9 - 7385*x^8 + 932*x^7 + 26931*x^6 + 14618*x^5 - 4894*x^4 + 334*x^3 + 2048*x^2 - 312*x + 41) 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 - 4*x^23 + 8*x^22 - 10*x^21 + 70*x^20 - 182*x^19 + 269*x^18 - 372*x^17 + 1647*x^16 - 1870*x^15 + 851*x^14 - 1270*x^13 + 4539*x^12 + 10226*x^11 - 6485*x^10 - 6130*x^9 - 7385*x^8 + 932*x^7 + 26931*x^6 + 14618*x^5 - 4894*x^4 + 334*x^3 + 2048*x^2 - 312*x + 41, 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 - 4*x^23 + 8*x^22 - 10*x^21 + 70*x^20 - 182*x^19 + 269*x^18 - 372*x^17 + 1647*x^16 - 1870*x^15 + 851*x^14 - 1270*x^13 + 4539*x^12 + 10226*x^11 - 6485*x^10 - 6130*x^9 - 7385*x^8 + 932*x^7 + 26931*x^6 + 14618*x^5 - 4894*x^4 + 334*x^3 + 2048*x^2 - 312*x + 41); 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 - 4*x^23 + 8*x^22 - 10*x^21 + 70*x^20 - 182*x^19 + 269*x^18 - 372*x^17 + 1647*x^16 - 1870*x^15 + 851*x^14 - 1270*x^13 + 4539*x^12 + 10226*x^11 - 6485*x^10 - 6130*x^9 - 7385*x^8 + 932*x^7 + 26931*x^6 + 14618*x^5 - 4894*x^4 + 334*x^3 + 2048*x^2 - 312*x + 41); 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

$C_3:D_8$ (as 24T37):

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 48
The 12 conjugacy class representatives for $C_3:D_8$
Character table for $C_3:D_8$

Intermediate fields

\(\Q(\sqrt{2}) \), 3.3.257.1, deg 4, 6.6.33817088.2, deg 8, deg 12

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 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 ${\href{/padicField/3.8.0.1}{8} }^{3}$ ${\href{/padicField/5.8.0.1}{8} }^{3}$ ${\href{/padicField/7.2.0.1}{2} }^{11}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{4}$ ${\href{/padicField/17.12.0.1}{12} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }^{3}$ ${\href{/padicField/23.12.0.1}{12} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{11}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{11}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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
\(2\) Copy content Toggle raw display 2.12.18.23$x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 7 x^{6} + 2 x^{5} + 8 x^{4} + 6 x^{3} + x^{2} + 6 x + 7$$2$$6$$18$$C_6\times C_2$$$[3]^{6}$$
2.12.18.23$x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 7 x^{6} + 2 x^{5} + 8 x^{4} + 6 x^{3} + x^{2} + 6 x + 7$$2$$6$$18$$C_6\times C_2$$$[3]^{6}$$
\(257\) Copy content Toggle raw display $\Q_{257}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{257}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$

Spectrum of ring of integers

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