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Properties

Label 40.0.130...936.2
Degree $40$
Signature $[0, 20]$
Discriminant $1.303\times 10^{71}$
Root discriminant \(59.96\)
Ramified primes $2,3,11$
Class number $5500$ (GRH)
Class group [5, 10, 110] (GRH)
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 4*x^38 + 15*x^36 - 56*x^34 + 209*x^32 - 780*x^30 + 2911*x^28 - 10864*x^26 + 40545*x^24 - 151316*x^22 + 564719*x^20 - 151316*x^18 + 40545*x^16 - 10864*x^14 + 2911*x^12 - 780*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1)
 
gp: K = bnfinit(y^40 - 4*y^38 + 15*y^36 - 56*y^34 + 209*y^32 - 780*y^30 + 2911*y^28 - 10864*y^26 + 40545*y^24 - 151316*y^22 + 564719*y^20 - 151316*y^18 + 40545*y^16 - 10864*y^14 + 2911*y^12 - 780*y^10 + 209*y^8 - 56*y^6 + 15*y^4 - 4*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 4*x^38 + 15*x^36 - 56*x^34 + 209*x^32 - 780*x^30 + 2911*x^28 - 10864*x^26 + 40545*x^24 - 151316*x^22 + 564719*x^20 - 151316*x^18 + 40545*x^16 - 10864*x^14 + 2911*x^12 - 780*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 4*x^38 + 15*x^36 - 56*x^34 + 209*x^32 - 780*x^30 + 2911*x^28 - 10864*x^26 + 40545*x^24 - 151316*x^22 + 564719*x^20 - 151316*x^18 + 40545*x^16 - 10864*x^14 + 2911*x^12 - 780*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1)
 

\( x^{40} - 4 x^{38} + 15 x^{36} - 56 x^{34} + 209 x^{32} - 780 x^{30} + 2911 x^{28} - 10864 x^{26} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $40$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 20]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(130305099804548492884220428175380349368393046678311823693003457545895936\) \(\medspace = 2^{80}\cdot 3^{20}\cdot 11^{36}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(59.96\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{2}3^{1/2}11^{9/10}\approx 59.96171354565991$
Ramified primes:   \(2\), \(3\), \(11\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $40$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(264=2^{3}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(259,·)$, $\chi_{264}(5,·)$, $\chi_{264}(263,·)$, $\chi_{264}(139,·)$, $\chi_{264}(145,·)$, $\chi_{264}(19,·)$, $\chi_{264}(149,·)$, $\chi_{264}(23,·)$, $\chi_{264}(25,·)$, $\chi_{264}(29,·)$, $\chi_{264}(163,·)$, $\chi_{264}(167,·)$, $\chi_{264}(169,·)$, $\chi_{264}(43,·)$, $\chi_{264}(173,·)$, $\chi_{264}(47,·)$, $\chi_{264}(49,·)$, $\chi_{264}(53,·)$, $\chi_{264}(191,·)$, $\chi_{264}(193,·)$, $\chi_{264}(67,·)$, $\chi_{264}(197,·)$, $\chi_{264}(71,·)$, $\chi_{264}(73,·)$, $\chi_{264}(211,·)$, $\chi_{264}(215,·)$, $\chi_{264}(217,·)$, $\chi_{264}(91,·)$, $\chi_{264}(221,·)$, $\chi_{264}(95,·)$, $\chi_{264}(97,·)$, $\chi_{264}(101,·)$, $\chi_{264}(235,·)$, $\chi_{264}(239,·)$, $\chi_{264}(241,·)$, $\chi_{264}(115,·)$, $\chi_{264}(245,·)$, $\chi_{264}(119,·)$, $\chi_{264}(125,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{564719}a^{22}-\frac{151316}{564719}$, $\frac{1}{564719}a^{23}-\frac{151316}{564719}a$, $\frac{1}{564719}a^{24}-\frac{151316}{564719}a^{2}$, $\frac{1}{564719}a^{25}-\frac{151316}{564719}a^{3}$, $\frac{1}{564719}a^{26}-\frac{151316}{564719}a^{4}$, $\frac{1}{564719}a^{27}-\frac{151316}{564719}a^{5}$, $\frac{1}{564719}a^{28}-\frac{151316}{564719}a^{6}$, $\frac{1}{564719}a^{29}-\frac{151316}{564719}a^{7}$, $\frac{1}{564719}a^{30}-\frac{151316}{564719}a^{8}$, $\frac{1}{564719}a^{31}-\frac{151316}{564719}a^{9}$, $\frac{1}{564719}a^{32}-\frac{151316}{564719}a^{10}$, $\frac{1}{564719}a^{33}-\frac{151316}{564719}a^{11}$, $\frac{1}{564719}a^{34}-\frac{151316}{564719}a^{12}$, $\frac{1}{564719}a^{35}-\frac{151316}{564719}a^{13}$, $\frac{1}{564719}a^{36}-\frac{151316}{564719}a^{14}$, $\frac{1}{564719}a^{37}-\frac{151316}{564719}a^{15}$, $\frac{1}{564719}a^{38}-\frac{151316}{564719}a^{16}$, $\frac{1}{564719}a^{39}-\frac{151316}{564719}a^{17}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{5}\times C_{10}\times C_{110}$, which has order $5500$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $19$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{209}{564719} a^{32} + \frac{408855776}{564719} a^{10} \)  (order $22$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{162180}{564719}a^{38}-\frac{605264}{564719}a^{36}+\frac{2270520}{564719}a^{34}-\frac{8473696}{564719}a^{32}+\frac{31625100}{564719}a^{30}-\frac{118026480}{564719}a^{28}+\frac{440480880}{564719}a^{26}-\frac{1643897025}{564719}a^{24}+\frac{6135107220}{564719}a^{22}-40545a^{20}+151316a^{18}-\frac{1643897025}{564719}a^{16}+\frac{6135107220}{564719}a^{14}-\frac{118026495}{564719}a^{12}+\frac{440480876}{564719}a^{10}-\frac{8473905}{564719}a^{8}+\frac{31625044}{564719}a^{6}-\frac{608175}{564719}a^{4}+\frac{162180}{564719}a^{2}-\frac{605264}{564719}$, $\frac{151316}{564719}a^{38}-\frac{605264}{564719}a^{36}+\frac{2269740}{564719}a^{34}-\frac{8473696}{564719}a^{32}+\frac{31625044}{564719}a^{30}-\frac{118026480}{564719}a^{28}+\frac{440480876}{564719}a^{26}-\frac{1643897025}{564719}a^{24}+\frac{6135107220}{564719}a^{22}-40545a^{20}+151316a^{18}-\frac{22896531856}{564719}a^{16}+\frac{6135107220}{564719}a^{14}-\frac{1643897024}{564719}a^{12}+\frac{440480876}{564719}a^{10}-\frac{118026480}{564719}a^{8}+\frac{31625044}{564719}a^{6}-\frac{8473696}{564719}a^{4}+\frac{162180}{564719}a^{2}-\frac{605264}{564719}$, $\frac{780}{564719}a^{34}-\frac{209}{564719}a^{32}+\frac{56}{564719}a^{30}+\frac{1525870529}{564719}a^{12}-\frac{408855776}{564719}a^{10}+\frac{109552575}{564719}a^{8}$, $\frac{780}{564719}a^{34}-\frac{1}{564719}a^{24}+\frac{1525870529}{564719}a^{12}-\frac{2107560}{564719}a^{2}$, $\frac{1}{564719}a^{25}+\frac{2107560}{564719}a^{3}$, $\frac{413403}{564719}a^{39}+\frac{10864}{564719}a^{38}-\frac{1653612}{564719}a^{37}+\frac{6201045}{564719}a^{35}-\frac{23150568}{564719}a^{33}+\frac{86401227}{564719}a^{31}-\frac{322454340}{564719}a^{29}+\frac{1203416133}{564719}a^{27}-\frac{4}{564719}a^{26}-\frac{4491210192}{564719}a^{25}+\frac{16761424635}{564719}a^{23}-110771a^{21}+413403a^{19}-\frac{62554488348}{564719}a^{17}+\frac{21252634831}{564719}a^{16}+\frac{16761424635}{564719}a^{15}-\frac{4491210192}{564719}a^{13}+\frac{1203416133}{564719}a^{11}-\frac{322454340}{564719}a^{9}+\frac{86401227}{564719}a^{7}-\frac{23150568}{564719}a^{5}-\frac{7865521}{564719}a^{4}+\frac{6201045}{564719}a^{3}-\frac{1653612}{564719}a$, $\frac{29681}{564719}a^{39}-\frac{56}{564719}a^{30}+\frac{4}{564719}a^{26}+\frac{58063278153}{564719}a^{17}-\frac{109552575}{564719}a^{8}+\frac{7865521}{564719}a^{4}$, $\frac{780}{564719}a^{34}-\frac{153}{564719}a^{31}+\frac{15}{564719}a^{28}+\frac{1525870529}{564719}a^{12}-\frac{299303201}{564719}a^{9}+\frac{29354524}{564719}a^{6}$, $\frac{29681}{564719}a^{39}+\frac{209}{564719}a^{32}-\frac{1}{564719}a^{24}+\frac{58063278153}{564719}a^{17}+\frac{408855776}{564719}a^{10}-\frac{2107560}{564719}a^{2}$, $\frac{4}{564719}a^{26}-\frac{3}{564719}a^{25}+\frac{1}{564719}a^{24}+\frac{7865521}{564719}a^{4}-\frac{5757961}{564719}a^{3}+\frac{2107560}{564719}a^{2}$, $\frac{10864}{564719}a^{38}-\frac{780}{564719}a^{35}+\frac{209}{564719}a^{32}+\frac{21252634831}{564719}a^{16}-\frac{1525870529}{564719}a^{13}+\frac{408855776}{564719}a^{10}$, $\frac{209}{564719}a^{32}+\frac{15}{564719}a^{29}+\frac{4}{564719}a^{26}+\frac{408855776}{564719}a^{10}+\frac{29354524}{564719}a^{7}+\frac{7865521}{564719}a^{4}$, $\frac{151316}{564719}a^{39}-\frac{151316}{564719}a^{38}-\frac{605264}{564719}a^{37}+\frac{605264}{564719}a^{36}+\frac{2269740}{564719}a^{35}-\frac{2269740}{564719}a^{34}-\frac{8473696}{564719}a^{33}+\frac{8473696}{564719}a^{32}+\frac{31625044}{564719}a^{31}-\frac{31625044}{564719}a^{30}-\frac{118026480}{564719}a^{29}+\frac{118026480}{564719}a^{28}+\frac{440480876}{564719}a^{27}-\frac{440480876}{564719}a^{26}-\frac{1643897024}{564719}a^{25}+\frac{1643897024}{564719}a^{24}+\frac{6135107220}{564719}a^{23}-\frac{6135107220}{564719}a^{22}-40545a^{21}+40545a^{20}+151316a^{19}-151316a^{18}-\frac{22896531856}{564719}a^{17}+\frac{22896531856}{564719}a^{16}+\frac{6135107220}{564719}a^{15}-\frac{6135107220}{564719}a^{14}-\frac{1643897024}{564719}a^{13}+\frac{1643897024}{564719}a^{12}+\frac{440480876}{564719}a^{11}-\frac{440480876}{564719}a^{10}-\frac{118026480}{564719}a^{9}+\frac{118026480}{564719}a^{8}+\frac{31625044}{564719}a^{7}-\frac{31625044}{564719}a^{6}-\frac{8473696}{564719}a^{5}+\frac{8473696}{564719}a^{4}+\frac{2269740}{564719}a^{3}-\frac{2269740}{564719}a^{2}-\frac{605264}{564719}a+\frac{1169983}{564719}$, $\frac{162180}{564719}a^{39}+\frac{313496}{564719}a^{38}-\frac{608175}{564719}a^{37}-\frac{1213439}{564719}a^{36}+\frac{2270520}{564719}a^{35}+\frac{4540260}{564719}a^{34}-\frac{8473905}{564719}a^{33}-\frac{16947601}{564719}a^{32}+\frac{31625100}{564719}a^{31}+\frac{63250144}{564719}a^{30}-\frac{118026495}{564719}a^{29}-\frac{236052975}{564719}a^{28}+\frac{440480880}{564719}a^{27}+\frac{880961756}{564719}a^{26}-\frac{1643897025}{564719}a^{25}-\frac{3287794049}{564719}a^{24}+\frac{6135107220}{564719}a^{23}+\frac{12270214440}{564719}a^{22}-40545a^{21}-81090a^{20}+151316a^{19}+302632a^{18}-\frac{1643897025}{564719}a^{17}-\frac{24540428881}{564719}a^{16}+\frac{440480880}{564719}a^{15}+\frac{6575588100}{564719}a^{14}-\frac{118026495}{564719}a^{13}-\frac{1761923519}{564719}a^{12}+\frac{31625100}{564719}a^{11}+\frac{472105976}{564719}a^{10}-\frac{8473905}{564719}a^{9}-\frac{126500385}{564719}a^{8}+\frac{2270520}{564719}a^{7}+\frac{33895564}{564719}a^{6}-\frac{608175}{564719}a^{5}-\frac{9081871}{564719}a^{4}+\frac{162180}{564719}a^{3}+\frac{2431920}{564719}a^{2}-\frac{40545}{564719}a-\frac{645809}{564719}$, $\frac{151316}{564719}a^{39}-\frac{10864}{564719}a^{38}-\frac{605264}{564719}a^{37}+\frac{2269740}{564719}a^{35}-\frac{8473696}{564719}a^{33}+\frac{31625044}{564719}a^{31}-\frac{118026480}{564719}a^{29}+\frac{440480876}{564719}a^{27}+\frac{4}{564719}a^{26}-\frac{1643897024}{564719}a^{25}+\frac{6135107220}{564719}a^{23}-40545a^{21}+151316a^{19}-\frac{22896531856}{564719}a^{17}-\frac{21252634831}{564719}a^{16}+\frac{6135107220}{564719}a^{15}-\frac{1643897024}{564719}a^{13}+\frac{440480876}{564719}a^{11}-\frac{118026480}{564719}a^{9}+\frac{31625044}{564719}a^{7}-\frac{8473696}{564719}a^{5}+\frac{7865521}{564719}a^{4}+\frac{2269740}{564719}a^{3}-\frac{605264}{564719}a$, $\frac{162180}{564719}a^{38}-\frac{608175}{564719}a^{36}+\frac{2270520}{564719}a^{34}-\frac{8473905}{564719}a^{32}+\frac{31625100}{564719}a^{30}-\frac{118026510}{564719}a^{28}+\frac{440480880}{564719}a^{26}-\frac{1643897025}{564719}a^{24}+\frac{6135107220}{564719}a^{22}-40545a^{20}+151316a^{18}-\frac{1643897025}{564719}a^{16}+\frac{440480880}{564719}a^{14}-\frac{118026495}{564719}a^{12}+\frac{31625100}{564719}a^{10}-\frac{8473905}{564719}a^{8}-\frac{27084004}{564719}a^{6}-\frac{608175}{564719}a^{4}+\frac{162180}{564719}a^{2}-a-\frac{40545}{564719}$, $\frac{151316}{564719}a^{39}+\frac{10864}{564719}a^{38}-\frac{600222}{564719}a^{37}-\frac{2131}{564719}a^{36}+\frac{2269740}{564719}a^{35}-\frac{8473696}{564719}a^{33}+\frac{31625044}{564719}a^{31}+\frac{41}{564719}a^{30}-\frac{118026506}{564719}a^{29}-\frac{15}{564719}a^{28}+\frac{440480876}{564719}a^{27}-\frac{1643897024}{564719}a^{25}+\frac{6135107220}{564719}a^{23}-\frac{1}{564719}a^{22}-40545a^{21}+151316a^{19}-\frac{22896531856}{564719}a^{17}+\frac{21252634831}{564719}a^{16}+\frac{15998489371}{564719}a^{15}-\frac{4168755811}{564719}a^{14}-\frac{1643897024}{564719}a^{13}+\frac{440480876}{564719}a^{11}-\frac{118026480}{564719}a^{9}+\frac{80198051}{564719}a^{8}-\frac{19218483}{564719}a^{7}-\frac{29354524}{564719}a^{6}-\frac{8473696}{564719}a^{5}+\frac{2269740}{564719}a^{3}-\frac{1169983}{564719}a+\frac{151316}{564719}$, $\frac{2911}{564719}a^{36}-\frac{209}{564719}a^{32}+\frac{1}{564719}a^{23}+\frac{5694626340}{564719}a^{14}-\frac{408855776}{564719}a^{10}+\frac{2107560}{564719}a$, $\frac{605264}{564719}a^{39}-\frac{2269740}{564719}a^{37}+\frac{8473696}{564719}a^{35}-\frac{31625044}{564719}a^{33}+\frac{209}{564719}a^{32}+\frac{118026480}{564719}a^{31}-\frac{440480876}{564719}a^{29}-\frac{15}{564719}a^{28}+\frac{1643897024}{564719}a^{27}-\frac{6135107220}{564719}a^{25}+\frac{22896531856}{564719}a^{23}-151316a^{21}+564719a^{19}-\frac{6135107220}{564719}a^{17}+\frac{1643897024}{564719}a^{15}-\frac{440480876}{564719}a^{13}+\frac{118026480}{564719}a^{11}+\frac{408855776}{564719}a^{10}-\frac{31625044}{564719}a^{9}+\frac{8473696}{564719}a^{7}-\frac{29354524}{564719}a^{6}-\frac{2269740}{564719}a^{5}+\frac{605264}{564719}a^{3}-\frac{151316}{564719}a$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 24039412784024376 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 24039412784024376 \cdot 5500}{22\cdot\sqrt{130305099804548492884220428175380349368393046678311823693003457545895936}}\cr\approx \mathstrut & 0.153101880167753 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^40 - 4*x^38 + 15*x^36 - 56*x^34 + 209*x^32 - 780*x^30 + 2911*x^28 - 10864*x^26 + 40545*x^24 - 151316*x^22 + 564719*x^20 - 151316*x^18 + 40545*x^16 - 10864*x^14 + 2911*x^12 - 780*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^40 - 4*x^38 + 15*x^36 - 56*x^34 + 209*x^32 - 780*x^30 + 2911*x^28 - 10864*x^26 + 40545*x^24 - 151316*x^22 + 564719*x^20 - 151316*x^18 + 40545*x^16 - 10864*x^14 + 2911*x^12 - 780*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^40 - 4*x^38 + 15*x^36 - 56*x^34 + 209*x^32 - 780*x^30 + 2911*x^28 - 10864*x^26 + 40545*x^24 - 151316*x^22 + 564719*x^20 - 151316*x^18 + 40545*x^16 - 10864*x^14 + 2911*x^12 - 780*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 4*x^38 + 15*x^36 - 56*x^34 + 209*x^32 - 780*x^30 + 2911*x^28 - 10864*x^26 + 40545*x^24 - 151316*x^22 + 564719*x^20 - 151316*x^18 + 40545*x^16 - 10864*x^14 + 2911*x^12 - 780*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1);
 
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

$C_2^2\times C_{10}$ (as 40T7):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{22}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{66}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-6}, \sqrt{22})\), \(\Q(\sqrt{3}, \sqrt{22})\), \(\Q(\sqrt{-2}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{-33})\), \(\Q(\sqrt{3}, \sqrt{-11})\), \(\Q(\sqrt{-6}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\zeta_{11})^+\), 8.0.77720518656.2, 10.10.77265229938688.1, 10.0.586732839846912.1, 10.0.1706859170463744.1, 10.10.18775450875101184.1, 10.10.53339349076992.1, 10.0.7024111812608.1, \(\Q(\zeta_{11})\), 20.0.360977976896857923653306611918700544.4, 20.20.360977976896857923653306611918700544.1, 20.0.5969915757478328440239161344.5, 20.0.360977976896857923653306611918700544.6, 20.0.344255425354822086003595935744.3, 20.0.352517555563337816067682238201856.8, 20.0.2983289065263288625233938941476864.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.10.0.1}{10} }^{4}$ ${\href{/padicField/7.10.0.1}{10} }^{4}$ R ${\href{/padicField/13.10.0.1}{10} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{20}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{20}$ ${\href{/padicField/47.10.0.1}{10} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{4}$ ${\href{/padicField/59.5.0.1}{5} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $40$$4$$10$$80$
\(3\) Copy content Toggle raw display 3.10.5.1$x^{10} + 162 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} + 162 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} + 162 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} + 162 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(11\) Copy content Toggle raw display 11.10.9.7$x^{10} + 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$