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Properties

Label 15.15.899...561.2
Degree $15$
Signature $[15, 0]$
Discriminant $8.996\times 10^{39}$
Root discriminant \(460.90\)
Ramified primes $17,97,131$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $A_8$ (as 15T72)

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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^15 - 3*x^14 - 424*x^13 + 549*x^12 + 65358*x^11 - 5743*x^10 - 4867029*x^9 - 4560961*x^8 + 186322787*x^7 + 356354865*x^6 - 3397215797*x^5 - 10179835125*x^4 + 18214687705*x^3 + 99761473375*x^2 + 121109770197*x + 42544769744)
 
Copy content gp:K = bnfinit(y^15 - 3*y^14 - 424*y^13 + 549*y^12 + 65358*y^11 - 5743*y^10 - 4867029*y^9 - 4560961*y^8 + 186322787*y^7 + 356354865*y^6 - 3397215797*y^5 - 10179835125*y^4 + 18214687705*y^3 + 99761473375*y^2 + 121109770197*y + 42544769744, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 3*x^14 - 424*x^13 + 549*x^12 + 65358*x^11 - 5743*x^10 - 4867029*x^9 - 4560961*x^8 + 186322787*x^7 + 356354865*x^6 - 3397215797*x^5 - 10179835125*x^4 + 18214687705*x^3 + 99761473375*x^2 + 121109770197*x + 42544769744);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 3*x^14 - 424*x^13 + 549*x^12 + 65358*x^11 - 5743*x^10 - 4867029*x^9 - 4560961*x^8 + 186322787*x^7 + 356354865*x^6 - 3397215797*x^5 - 10179835125*x^4 + 18214687705*x^3 + 99761473375*x^2 + 121109770197*x + 42544769744)
 

\( x^{15} - 3 x^{14} - 424 x^{13} + 549 x^{12} + 65358 x^{11} - 5743 x^{10} - 4867029 x^{9} + \cdots + 42544769744 \) 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:  $15$
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:  $[15, 0]$
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:   \(8995776251690372238394631846990504365561\) \(\medspace = 17^{6}\cdot 97^{10}\cdot 131^{6}\) 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:  \(460.90\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $17^{1/2}97^{2/3}131^{1/2}\approx 996.264368545655$
Ramified primes:   \(17\), \(97\), \(131\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{121}a^{13}-\frac{4}{121}a^{12}-\frac{27}{121}a^{11}-\frac{28}{121}a^{10}-\frac{38}{121}a^{9}-\frac{1}{11}a^{8}+\frac{35}{121}a^{7}+\frac{1}{11}a^{6}+\frac{40}{121}a^{5}-\frac{9}{121}a^{4}+\frac{46}{121}a^{3}-\frac{50}{121}a^{2}-\frac{6}{121}a+\frac{27}{121}$, $\frac{1}{60\cdots 01}a^{14}-\frac{22\cdots 06}{60\cdots 01}a^{13}-\frac{13\cdots 57}{60\cdots 01}a^{12}+\frac{26\cdots 19}{60\cdots 01}a^{11}+\frac{63\cdots 97}{60\cdots 01}a^{10}+\frac{29\cdots 95}{60\cdots 01}a^{9}+\frac{37\cdots 62}{60\cdots 01}a^{8}-\frac{19\cdots 63}{60\cdots 01}a^{7}+\frac{11\cdots 32}{60\cdots 01}a^{6}-\frac{20\cdots 81}{54\cdots 91}a^{5}-\frac{25\cdots 19}{60\cdots 01}a^{4}+\frac{24\cdots 44}{60\cdots 01}a^{3}-\frac{30\cdots 67}{60\cdots 01}a^{2}+\frac{28\cdots 11}{60\cdots 01}a+\frac{22\cdots 19}{60\cdots 01}$ 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:  $C_{3}$, which has order $3$ (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:  $C_{2}\times C_{2}\times C_{6}$, which has order $24$ (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:   $\frac{54\cdots 52}{60\cdots 01}a^{14}-\frac{60\cdots 41}{60\cdots 01}a^{13}-\frac{18\cdots 08}{60\cdots 01}a^{12}+\frac{18\cdots 20}{60\cdots 01}a^{11}+\frac{22\cdots 49}{60\cdots 01}a^{10}-\frac{18\cdots 07}{60\cdots 01}a^{9}-\frac{14\cdots 87}{60\cdots 01}a^{8}+\frac{80\cdots 49}{60\cdots 01}a^{7}+\frac{52\cdots 08}{60\cdots 01}a^{6}-\frac{13\cdots 35}{54\cdots 91}a^{5}-\frac{10\cdots 69}{60\cdots 01}a^{4}+\frac{28\cdots 25}{60\cdots 01}a^{3}+\frac{85\cdots 92}{60\cdots 01}a^{2}+\frac{13\cdots 69}{60\cdots 01}a+\frac{56\cdots 39}{60\cdots 01}$, $\frac{71\cdots 99}{60\cdots 01}a^{14}+\frac{32\cdots 90}{60\cdots 01}a^{13}-\frac{57\cdots 95}{60\cdots 01}a^{12}-\frac{10\cdots 94}{60\cdots 01}a^{11}+\frac{11\cdots 61}{60\cdots 01}a^{10}+\frac{11\cdots 04}{60\cdots 01}a^{9}-\frac{89\cdots 11}{60\cdots 01}a^{8}-\frac{52\cdots 60}{60\cdots 01}a^{7}+\frac{31\cdots 46}{60\cdots 01}a^{6}+\frac{11\cdots 84}{54\cdots 91}a^{5}-\frac{51\cdots 97}{60\cdots 01}a^{4}-\frac{17\cdots 43}{60\cdots 01}a^{3}+\frac{28\cdots 76}{60\cdots 01}a^{2}+\frac{95\cdots 90}{60\cdots 01}a+\frac{45\cdots 47}{60\cdots 01}$, $\frac{26\cdots 89}{60\cdots 01}a^{14}+\frac{93\cdots 66}{60\cdots 01}a^{13}-\frac{11\cdots 48}{60\cdots 01}a^{12}-\frac{55\cdots 22}{60\cdots 01}a^{11}+\frac{15\cdots 58}{60\cdots 01}a^{10}+\frac{91\cdots 81}{60\cdots 01}a^{9}-\frac{83\cdots 02}{60\cdots 01}a^{8}-\frac{64\cdots 88}{60\cdots 01}a^{7}+\frac{15\cdots 79}{60\cdots 01}a^{6}+\frac{17\cdots 52}{54\cdots 91}a^{5}+\frac{18\cdots 93}{60\cdots 01}a^{4}-\frac{19\cdots 42}{60\cdots 01}a^{3}-\frac{62\cdots 49}{60\cdots 01}a^{2}-\frac{68\cdots 31}{60\cdots 01}a-\frac{24\cdots 37}{60\cdots 01}$, $\frac{20\cdots 35}{60\cdots 01}a^{14}-\frac{68\cdots 76}{60\cdots 01}a^{13}-\frac{90\cdots 47}{60\cdots 01}a^{12}+\frac{16\cdots 82}{60\cdots 01}a^{11}+\frac{14\cdots 49}{60\cdots 01}a^{10}-\frac{14\cdots 80}{60\cdots 01}a^{9}-\frac{10\cdots 16}{60\cdots 01}a^{8}+\frac{25\cdots 70}{60\cdots 01}a^{7}+\frac{42\cdots 83}{60\cdots 01}a^{6}+\frac{20\cdots 86}{54\cdots 91}a^{5}-\frac{81\cdots 95}{60\cdots 01}a^{4}-\frac{10\cdots 07}{60\cdots 01}a^{3}+\frac{58\cdots 03}{60\cdots 01}a^{2}+\frac{13\cdots 95}{60\cdots 01}a+\frac{57\cdots 41}{60\cdots 01}$, $\frac{79\cdots 17}{60\cdots 01}a^{14}-\frac{28\cdots 77}{60\cdots 01}a^{13}-\frac{33\cdots 77}{60\cdots 01}a^{12}+\frac{64\cdots 08}{60\cdots 01}a^{11}+\frac{51\cdots 75}{60\cdots 01}a^{10}-\frac{37\cdots 43}{60\cdots 01}a^{9}-\frac{38\cdots 24}{60\cdots 01}a^{8}-\frac{10\cdots 98}{60\cdots 01}a^{7}+\frac{14\cdots 55}{60\cdots 01}a^{6}+\frac{16\cdots 73}{54\cdots 91}a^{5}-\frac{28\cdots 73}{60\cdots 01}a^{4}-\frac{61\cdots 79}{60\cdots 01}a^{3}+\frac{18\cdots 03}{60\cdots 01}a^{2}+\frac{66\cdots 10}{60\cdots 01}a+\frac{51\cdots 09}{60\cdots 01}$, $\frac{24\cdots 01}{60\cdots 01}a^{14}-\frac{87\cdots 03}{60\cdots 01}a^{13}-\frac{10\cdots 29}{60\cdots 01}a^{12}+\frac{19\cdots 64}{60\cdots 01}a^{11}+\frac{15\cdots 56}{60\cdots 01}a^{10}-\frac{10\cdots 62}{60\cdots 01}a^{9}-\frac{11\cdots 61}{60\cdots 01}a^{8}-\frac{38\cdots 94}{60\cdots 01}a^{7}+\frac{45\cdots 77}{60\cdots 01}a^{6}+\frac{52\cdots 28}{54\cdots 91}a^{5}-\frac{85\cdots 89}{60\cdots 01}a^{4}-\frac{19\cdots 82}{60\cdots 01}a^{3}+\frac{56\cdots 18}{60\cdots 01}a^{2}+\frac{20\cdots 02}{60\cdots 01}a+\frac{16\cdots 71}{60\cdots 01}$, $\frac{16\cdots 49}{60\cdots 01}a^{14}-\frac{11\cdots 03}{60\cdots 01}a^{13}-\frac{66\cdots 39}{60\cdots 01}a^{12}+\frac{32\cdots 13}{60\cdots 01}a^{11}+\frac{92\cdots 49}{60\cdots 01}a^{10}-\frac{33\cdots 54}{60\cdots 01}a^{9}-\frac{60\cdots 78}{60\cdots 01}a^{8}+\frac{14\cdots 96}{60\cdots 01}a^{7}+\frac{20\cdots 44}{60\cdots 01}a^{6}-\frac{21\cdots 93}{54\cdots 91}a^{5}-\frac{34\cdots 24}{60\cdots 01}a^{4}-\frac{81\cdots 77}{60\cdots 01}a^{3}+\frac{21\cdots 43}{60\cdots 01}a^{2}+\frac{34\cdots 51}{60\cdots 01}a+\frac{13\cdots 95}{60\cdots 01}$, $\frac{20\cdots 75}{54\cdots 91}a^{14}-\frac{77\cdots 19}{54\cdots 91}a^{13}-\frac{89\cdots 03}{54\cdots 91}a^{12}+\frac{21\cdots 60}{54\cdots 91}a^{11}+\frac{14\cdots 70}{54\cdots 91}a^{10}-\frac{21\cdots 27}{54\cdots 91}a^{9}-\frac{10\cdots 94}{54\cdots 91}a^{8}+\frac{82\cdots 86}{54\cdots 91}a^{7}+\frac{43\cdots 54}{54\cdots 91}a^{6}+\frac{42\cdots 20}{49\cdots 81}a^{5}-\frac{88\cdots 86}{54\cdots 91}a^{4}-\frac{92\cdots 47}{54\cdots 91}a^{3}+\frac{68\cdots 15}{54\cdots 91}a^{2}+\frac{14\cdots 34}{54\cdots 91}a+\frac{61\cdots 75}{54\cdots 91}$, $\frac{26\cdots 40}{60\cdots 01}a^{14}-\frac{25\cdots 55}{60\cdots 01}a^{13}-\frac{94\cdots 94}{60\cdots 01}a^{12}+\frac{74\cdots 88}{60\cdots 01}a^{11}+\frac{11\cdots 02}{60\cdots 01}a^{10}-\frac{71\cdots 98}{60\cdots 01}a^{9}-\frac{75\cdots 42}{60\cdots 01}a^{8}+\frac{28\cdots 88}{60\cdots 01}a^{7}+\frac{26\cdots 26}{60\cdots 01}a^{6}-\frac{37\cdots 43}{54\cdots 91}a^{5}-\frac{49\cdots 92}{60\cdots 01}a^{4}-\frac{22\cdots 57}{60\cdots 01}a^{3}+\frac{35\cdots 93}{60\cdots 01}a^{2}+\frac{77\cdots 55}{60\cdots 01}a+\frac{47\cdots 55}{60\cdots 01}$, $\frac{34\cdots 08}{60\cdots 01}a^{14}-\frac{39\cdots 05}{60\cdots 01}a^{13}-\frac{96\cdots 33}{60\cdots 01}a^{12}+\frac{10\cdots 88}{60\cdots 01}a^{11}+\frac{70\cdots 95}{60\cdots 01}a^{10}-\frac{83\cdots 83}{60\cdots 01}a^{9}-\frac{34\cdots 14}{60\cdots 01}a^{8}+\frac{26\cdots 58}{60\cdots 01}a^{7}-\frac{14\cdots 26}{60\cdots 01}a^{6}-\frac{45\cdots 20}{54\cdots 91}a^{5}+\frac{55\cdots 10}{60\cdots 01}a^{4}+\frac{12\cdots 58}{60\cdots 01}a^{3}-\frac{58\cdots 46}{60\cdots 01}a^{2}-\frac{20\cdots 09}{60\cdots 01}a-\frac{16\cdots 03}{60\cdots 01}$, $\frac{17\cdots 31}{60\cdots 01}a^{14}-\frac{12\cdots 67}{60\cdots 01}a^{13}-\frac{68\cdots 49}{60\cdots 01}a^{12}+\frac{36\cdots 88}{60\cdots 01}a^{11}+\frac{98\cdots 36}{60\cdots 01}a^{10}-\frac{39\cdots 73}{60\cdots 01}a^{9}-\frac{67\cdots 58}{60\cdots 01}a^{8}+\frac{18\cdots 14}{60\cdots 01}a^{7}+\frac{24\cdots 65}{60\cdots 01}a^{6}-\frac{28\cdots 70}{54\cdots 91}a^{5}-\frac{44\cdots 20}{60\cdots 01}a^{4}-\frac{91\cdots 01}{60\cdots 01}a^{3}+\frac{31\cdots 31}{60\cdots 01}a^{2}+\frac{53\cdots 61}{60\cdots 01}a+\frac{21\cdots 03}{60\cdots 01}$, $\frac{26\cdots 18}{60\cdots 01}a^{14}-\frac{29\cdots 49}{60\cdots 01}a^{13}-\frac{89\cdots 08}{60\cdots 01}a^{12}+\frac{87\cdots 77}{60\cdots 01}a^{11}+\frac{10\cdots 91}{60\cdots 01}a^{10}-\frac{85\cdots 02}{60\cdots 01}a^{9}-\frac{60\cdots 26}{60\cdots 01}a^{8}+\frac{36\cdots 36}{60\cdots 01}a^{7}+\frac{20\cdots 54}{60\cdots 01}a^{6}-\frac{60\cdots 86}{54\cdots 91}a^{5}-\frac{36\cdots 05}{60\cdots 01}a^{4}+\frac{24\cdots 75}{60\cdots 01}a^{3}+\frac{28\cdots 49}{60\cdots 01}a^{2}+\frac{38\cdots 87}{60\cdots 01}a+\frac{14\cdots 33}{60\cdots 01}$, $\frac{21\cdots 48}{60\cdots 01}a^{14}+\frac{63\cdots 41}{60\cdots 01}a^{13}-\frac{75\cdots 21}{60\cdots 01}a^{12}-\frac{27\cdots 59}{60\cdots 01}a^{11}+\frac{30\cdots 65}{60\cdots 01}a^{10}+\frac{38\cdots 23}{60\cdots 01}a^{9}+\frac{91\cdots 18}{60\cdots 01}a^{8}-\frac{23\cdots 59}{60\cdots 01}a^{7}-\frac{10\cdots 95}{60\cdots 01}a^{6}+\frac{50\cdots 63}{54\cdots 91}a^{5}+\frac{38\cdots 04}{60\cdots 01}a^{4}-\frac{55\cdots 27}{60\cdots 01}a^{3}-\frac{44\cdots 16}{60\cdots 01}a^{2}-\frac{10\cdots 09}{60\cdots 01}a-\frac{68\cdots 19}{60\cdots 01}$, $\frac{16\cdots 17}{60\cdots 01}a^{14}-\frac{80\cdots 66}{60\cdots 01}a^{13}-\frac{67\cdots 20}{60\cdots 01}a^{12}+\frac{21\cdots 10}{60\cdots 01}a^{11}+\frac{10\cdots 41}{60\cdots 01}a^{10}-\frac{19\cdots 71}{60\cdots 01}a^{9}-\frac{73\cdots 88}{60\cdots 01}a^{8}+\frac{62\cdots 07}{60\cdots 01}a^{7}+\frac{27\cdots 49}{60\cdots 01}a^{6}+\frac{45\cdots 12}{54\cdots 91}a^{5}-\frac{51\cdots 13}{60\cdots 01}a^{4}-\frac{59\cdots 54}{60\cdots 01}a^{3}+\frac{36\cdots 08}{60\cdots 01}a^{2}+\frac{77\cdots 51}{60\cdots 01}a+\frac{33\cdots 09}{60\cdots 01}$ 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:  \( 2167432538320000 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 2167432538320000 \cdot 3}{2\cdot\sqrt{8995776251690372238394631846990504365561}}\cr\approx \mathstrut & 1.12322680867629 \end{aligned}\] (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^15 - 3*x^14 - 424*x^13 + 549*x^12 + 65358*x^11 - 5743*x^10 - 4867029*x^9 - 4560961*x^8 + 186322787*x^7 + 356354865*x^6 - 3397215797*x^5 - 10179835125*x^4 + 18214687705*x^3 + 99761473375*x^2 + 121109770197*x + 42544769744) 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^15 - 3*x^14 - 424*x^13 + 549*x^12 + 65358*x^11 - 5743*x^10 - 4867029*x^9 - 4560961*x^8 + 186322787*x^7 + 356354865*x^6 - 3397215797*x^5 - 10179835125*x^4 + 18214687705*x^3 + 99761473375*x^2 + 121109770197*x + 42544769744, 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^15 - 3*x^14 - 424*x^13 + 549*x^12 + 65358*x^11 - 5743*x^10 - 4867029*x^9 - 4560961*x^8 + 186322787*x^7 + 356354865*x^6 - 3397215797*x^5 - 10179835125*x^4 + 18214687705*x^3 + 99761473375*x^2 + 121109770197*x + 42544769744); 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 = PolynomialRing(QQ); K, a = NumberField(x^15 - 3*x^14 - 424*x^13 + 549*x^12 + 65358*x^11 - 5743*x^10 - 4867029*x^9 - 4560961*x^8 + 186322787*x^7 + 356354865*x^6 - 3397215797*x^5 - 10179835125*x^4 + 18214687705*x^3 + 99761473375*x^2 + 121109770197*x + 42544769744); 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

$A_8$ (as 15T72):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 20160
The 14 conjugacy class representatives for $A_8$
Character table for $A_8$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 8 sibling: 8.8.46664208361.1
Degree 28 sibling: deg 28
Degree 35 sibling: deg 35
Arithmetically equivalent sibling: 15.15.8995776251690372238394631846990504365561.1
Minimal sibling: 8.8.46664208361.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.7.0.1}{7} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.7.0.1}{7} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.7.0.1}{7} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.7.0.1}{7} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ R ${\href{/padicField/19.7.0.1}{7} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ $15$ $15$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.7.0.1}{7} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.7.0.1}{7} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ $15$ ${\href{/padicField/59.7.0.1}{7} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(17\) Copy content Toggle raw display $\Q_{17}$$x + 14$$1$$1$$0$Trivial$$[\ ]$$
17.1.2.1a1.1$x^{2} + 17$$2$$1$$1$$C_2$$$[\ ]_{2}$$
17.1.2.1a1.2$x^{2} + 51$$2$$1$$1$$C_2$$$[\ ]_{2}$$
17.2.1.0a1.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
17.4.2.4a1.2$x^{8} + 14 x^{6} + 20 x^{5} + 55 x^{4} + 140 x^{3} + 142 x^{2} + 60 x + 26$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(97\) Copy content Toggle raw display 97.5.3.10a1.1$x^{15} + 9 x^{11} + 276 x^{10} + 27 x^{7} + 1656 x^{6} + 25392 x^{5} + 27 x^{3} + 2581 x^{2} + 76176 x + 778688$$3$$5$$10$$C_{15}$$$[\ ]_{3}^{5}$$
\(131\) Copy content Toggle raw display $\Q_{131}$$x + 129$$1$$1$$0$Trivial$$[\ ]$$
131.2.1.0a1.1$x^{2} + 127 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
131.2.2.2a1.1$x^{4} + 254 x^{3} + 16133 x^{2} + 639 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
131.4.2.4a1.2$x^{8} + 18 x^{6} + 218 x^{5} + 85 x^{4} + 1962 x^{3} + 11917 x^{2} + 436 x + 135$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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