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Properties

Label 44.44.490...521.1
Degree $44$
Signature $[44, 0]$
Discriminant $4.909\times 10^{107}$
Root discriminant \(280.23\)
Ramified primes $23,41$
Class number not computed
Class group not computed
Galois group $C_{44}$ (as 44T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 7*x^43 - 187*x^42 + 1072*x^41 + 17163*x^40 - 70103*x^39 - 997971*x^38 + 2375465*x^37 + 39641205*x^36 - 33307763*x^35 - 1096561800*x^34 - 533702618*x^33 + 20931922170*x^32 + 35314456183*x^31 - 264474855314*x^30 - 793926535475*x^29 + 1942918254504*x^28 + 10265625159992*x^27 - 3458448525180*x^26 - 80753578257977*x^25 - 78276745999632*x^24 + 358613104785724*x^23 + 809785514922531*x^22 - 568771257825877*x^21 - 3562658275456962*x^20 - 2066339006089008*x^19 + 7248452558678380*x^18 + 11554435372453566*x^17 - 3203124978259608*x^16 - 20843198648583718*x^15 - 11861788440019640*x^14 + 14310229227828665*x^13 + 19835277513240846*x^12 + 991200223617035*x^11 - 11320902849209936*x^10 - 5711158023758032*x^9 + 1999942105903247*x^8 + 2439911320594527*x^7 + 307599738065272*x^6 - 336467669357964*x^5 - 115847184197020*x^4 + 5127635333867*x^3 + 5921101695236*x^2 + 200575150497*x - 84271515023)
 
gp: K = bnfinit(y^44 - 7*y^43 - 187*y^42 + 1072*y^41 + 17163*y^40 - 70103*y^39 - 997971*y^38 + 2375465*y^37 + 39641205*y^36 - 33307763*y^35 - 1096561800*y^34 - 533702618*y^33 + 20931922170*y^32 + 35314456183*y^31 - 264474855314*y^30 - 793926535475*y^29 + 1942918254504*y^28 + 10265625159992*y^27 - 3458448525180*y^26 - 80753578257977*y^25 - 78276745999632*y^24 + 358613104785724*y^23 + 809785514922531*y^22 - 568771257825877*y^21 - 3562658275456962*y^20 - 2066339006089008*y^19 + 7248452558678380*y^18 + 11554435372453566*y^17 - 3203124978259608*y^16 - 20843198648583718*y^15 - 11861788440019640*y^14 + 14310229227828665*y^13 + 19835277513240846*y^12 + 991200223617035*y^11 - 11320902849209936*y^10 - 5711158023758032*y^9 + 1999942105903247*y^8 + 2439911320594527*y^7 + 307599738065272*y^6 - 336467669357964*y^5 - 115847184197020*y^4 + 5127635333867*y^3 + 5921101695236*y^2 + 200575150497*y - 84271515023, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - 7*x^43 - 187*x^42 + 1072*x^41 + 17163*x^40 - 70103*x^39 - 997971*x^38 + 2375465*x^37 + 39641205*x^36 - 33307763*x^35 - 1096561800*x^34 - 533702618*x^33 + 20931922170*x^32 + 35314456183*x^31 - 264474855314*x^30 - 793926535475*x^29 + 1942918254504*x^28 + 10265625159992*x^27 - 3458448525180*x^26 - 80753578257977*x^25 - 78276745999632*x^24 + 358613104785724*x^23 + 809785514922531*x^22 - 568771257825877*x^21 - 3562658275456962*x^20 - 2066339006089008*x^19 + 7248452558678380*x^18 + 11554435372453566*x^17 - 3203124978259608*x^16 - 20843198648583718*x^15 - 11861788440019640*x^14 + 14310229227828665*x^13 + 19835277513240846*x^12 + 991200223617035*x^11 - 11320902849209936*x^10 - 5711158023758032*x^9 + 1999942105903247*x^8 + 2439911320594527*x^7 + 307599738065272*x^6 - 336467669357964*x^5 - 115847184197020*x^4 + 5127635333867*x^3 + 5921101695236*x^2 + 200575150497*x - 84271515023);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - 7*x^43 - 187*x^42 + 1072*x^41 + 17163*x^40 - 70103*x^39 - 997971*x^38 + 2375465*x^37 + 39641205*x^36 - 33307763*x^35 - 1096561800*x^34 - 533702618*x^33 + 20931922170*x^32 + 35314456183*x^31 - 264474855314*x^30 - 793926535475*x^29 + 1942918254504*x^28 + 10265625159992*x^27 - 3458448525180*x^26 - 80753578257977*x^25 - 78276745999632*x^24 + 358613104785724*x^23 + 809785514922531*x^22 - 568771257825877*x^21 - 3562658275456962*x^20 - 2066339006089008*x^19 + 7248452558678380*x^18 + 11554435372453566*x^17 - 3203124978259608*x^16 - 20843198648583718*x^15 - 11861788440019640*x^14 + 14310229227828665*x^13 + 19835277513240846*x^12 + 991200223617035*x^11 - 11320902849209936*x^10 - 5711158023758032*x^9 + 1999942105903247*x^8 + 2439911320594527*x^7 + 307599738065272*x^6 - 336467669357964*x^5 - 115847184197020*x^4 + 5127635333867*x^3 + 5921101695236*x^2 + 200575150497*x - 84271515023)
 

\( x^{44} - 7 x^{43} - 187 x^{42} + 1072 x^{41} + 17163 x^{40} - 70103 x^{39} - 997971 x^{38} + \cdots - 84271515023 \) Copy content Toggle raw display

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[44, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(490\!\cdots\!521\) \(\medspace = 23^{40}\cdot 41^{33}\) 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:  \(280.23\)
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:  $23^{10/11}41^{3/4}\approx 280.2349440528612$
Ramified primes:   \(23\), \(41\) 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:  \(\Q(\sqrt{41}) \)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(943=23\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{943}(1,·)$, $\chi_{943}(901,·)$, $\chi_{943}(647,·)$, $\chi_{943}(9,·)$, $\chi_{943}(524,·)$, $\chi_{943}(450,·)$, $\chi_{943}(657,·)$, $\chi_{943}(532,·)$, $\chi_{943}(278,·)$, $\chi_{943}(409,·)$, $\chi_{943}(542,·)$, $\chi_{943}(32,·)$, $\chi_{943}(163,·)$, $\chi_{943}(165,·)$, $\chi_{943}(811,·)$, $\chi_{943}(173,·)$, $\chi_{943}(50,·)$, $\chi_{943}(565,·)$, $\chi_{943}(696,·)$, $\chi_{943}(698,·)$, $\chi_{943}(829,·)$, $\chi_{943}(821,·)$, $\chi_{943}(288,·)$, $\chi_{943}(706,·)$, $\chi_{943}(196,·)$, $\chi_{943}(583,·)$, $\chi_{943}(73,·)$, $\chi_{943}(81,·)$, $\chi_{943}(852,·)$, $\chi_{943}(729,·)$, $\chi_{943}(860,·)$, $\chi_{943}(606,·)$, $\chi_{943}(737,·)$, $\chi_{943}(739,·)$, $\chi_{943}(614,·)$, $\chi_{943}(616,·)$, $\chi_{943}(491,·)$, $\chi_{943}(624,·)$, $\chi_{943}(370,·)$, $\chi_{943}(903,·)$, $\chi_{943}(501,·)$, $\chi_{943}(788,·)$, $\chi_{943}(124,·)$, $\chi_{943}(255,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{2}a^{33}-\frac{1}{2}a^{31}-\frac{1}{2}a^{29}-\frac{1}{2}a^{28}-\frac{1}{2}a^{26}-\frac{1}{2}a^{24}-\frac{1}{2}a^{23}-\frac{1}{2}a^{21}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{34}-\frac{1}{2}a^{32}-\frac{1}{2}a^{30}-\frac{1}{2}a^{29}-\frac{1}{2}a^{27}-\frac{1}{2}a^{25}-\frac{1}{2}a^{24}-\frac{1}{2}a^{22}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{35}-\frac{1}{2}a^{30}-\frac{1}{2}a^{29}-\frac{1}{2}a^{25}-\frac{1}{2}a^{24}-\frac{1}{2}a^{21}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{36}-\frac{1}{2}a^{31}-\frac{1}{2}a^{30}-\frac{1}{2}a^{26}-\frac{1}{2}a^{25}-\frac{1}{2}a^{22}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{37}-\frac{1}{2}a^{32}-\frac{1}{2}a^{31}-\frac{1}{2}a^{27}-\frac{1}{2}a^{26}-\frac{1}{2}a^{23}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{38}-\frac{1}{2}a^{32}-\frac{1}{2}a^{31}-\frac{1}{2}a^{29}-\frac{1}{2}a^{27}-\frac{1}{2}a^{26}-\frac{1}{2}a^{23}-\frac{1}{2}a^{22}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{39}-\frac{1}{2}a^{32}-\frac{1}{2}a^{31}-\frac{1}{2}a^{30}-\frac{1}{2}a^{29}-\frac{1}{2}a^{27}-\frac{1}{2}a^{26}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{40}-\frac{1}{2}a^{32}-\frac{1}{2}a^{30}-\frac{1}{2}a^{29}-\frac{1}{2}a^{27}-\frac{1}{2}a^{26}-\frac{1}{2}a^{24}-\frac{1}{2}a^{23}-\frac{1}{2}a^{20}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{77006}a^{41}+\frac{902}{38503}a^{40}-\frac{7287}{77006}a^{39}+\frac{8685}{77006}a^{38}+\frac{3407}{77006}a^{37}+\frac{15613}{77006}a^{36}-\frac{74}{38503}a^{35}+\frac{11881}{77006}a^{34}+\frac{18817}{77006}a^{33}+\frac{17679}{38503}a^{32}+\frac{26041}{77006}a^{31}-\frac{3891}{38503}a^{30}+\frac{30437}{77006}a^{29}+\frac{20743}{77006}a^{28}+\frac{31143}{77006}a^{27}+\frac{16222}{38503}a^{26}-\frac{14951}{77006}a^{25}-\frac{292}{38503}a^{24}+\frac{5903}{38503}a^{23}-\frac{22285}{77006}a^{22}-\frac{12327}{38503}a^{21}-\frac{4538}{38503}a^{20}+\frac{25931}{77006}a^{19}-\frac{22543}{77006}a^{18}+\frac{18545}{38503}a^{17}+\frac{1749}{38503}a^{16}+\frac{37067}{77006}a^{15}+\frac{4650}{38503}a^{14}-\frac{5226}{38503}a^{13}-\frac{5680}{38503}a^{12}-\frac{32129}{77006}a^{11}+\frac{15624}{38503}a^{10}+\frac{6805}{77006}a^{9}-\frac{24873}{77006}a^{8}+\frac{14079}{38503}a^{7}-\frac{12437}{38503}a^{6}+\frac{4559}{38503}a^{5}-\frac{24657}{77006}a^{4}+\frac{27207}{77006}a^{3}+\frac{3451}{38503}a^{2}-\frac{3735}{38503}a+\frac{2027}{38503}$, $\frac{1}{77006}a^{42}+\frac{5526}{38503}a^{40}-\frac{13593}{77006}a^{39}+\frac{3194}{38503}a^{38}-\frac{4319}{38503}a^{37}+\frac{9098}{38503}a^{36}+\frac{4676}{38503}a^{35}-\frac{6839}{77006}a^{34}+\frac{10633}{77006}a^{33}-\frac{18663}{38503}a^{32}+\frac{26417}{77006}a^{31}-\frac{22933}{77006}a^{30}+\frac{17673}{77006}a^{29}-\frac{1408}{38503}a^{28}+\frac{26349}{77006}a^{27}-\frac{19367}{77006}a^{26}-\frac{19583}{77006}a^{25}+\frac{93}{278}a^{24}+\frac{10353}{77006}a^{23}+\frac{18857}{77006}a^{22}-\frac{4225}{77006}a^{21}+\frac{17630}{38503}a^{20}-\frac{10461}{38503}a^{19}-\frac{15756}{38503}a^{18}+\frac{5676}{38503}a^{17}+\frac{1332}{38503}a^{16}+\frac{20143}{77006}a^{15}-\frac{172}{38503}a^{14}+\frac{16081}{77006}a^{13}+\frac{8109}{38503}a^{12}-\frac{32057}{77006}a^{11}+\frac{3805}{77006}a^{10}+\frac{19867}{77006}a^{9}+\frac{2276}{38503}a^{8}-\frac{36449}{77006}a^{7}-\frac{6342}{38503}a^{6}+\frac{5755}{77006}a^{5}-\frac{1033}{77006}a^{4}+\frac{16799}{77006}a^{3}+\frac{8147}{38503}a^{2}-\frac{34619}{77006}a+\frac{1077}{38503}$, $\frac{1}{24\!\cdots\!46}a^{43}-\frac{20\!\cdots\!24}{12\!\cdots\!23}a^{42}-\frac{74\!\cdots\!40}{12\!\cdots\!23}a^{41}+\frac{23\!\cdots\!13}{24\!\cdots\!46}a^{40}-\frac{17\!\cdots\!99}{12\!\cdots\!23}a^{39}-\frac{28\!\cdots\!83}{12\!\cdots\!23}a^{38}-\frac{14\!\cdots\!15}{12\!\cdots\!23}a^{37}-\frac{46\!\cdots\!89}{24\!\cdots\!46}a^{36}+\frac{44\!\cdots\!17}{24\!\cdots\!46}a^{35}+\frac{39\!\cdots\!01}{24\!\cdots\!46}a^{34}+\frac{56\!\cdots\!82}{12\!\cdots\!23}a^{33}+\frac{28\!\cdots\!31}{24\!\cdots\!46}a^{32}+\frac{46\!\cdots\!37}{12\!\cdots\!23}a^{31}+\frac{26\!\cdots\!19}{12\!\cdots\!23}a^{30}-\frac{99\!\cdots\!51}{12\!\cdots\!23}a^{29}+\frac{55\!\cdots\!17}{24\!\cdots\!46}a^{28}+\frac{68\!\cdots\!61}{24\!\cdots\!46}a^{27}-\frac{78\!\cdots\!93}{12\!\cdots\!23}a^{26}-\frac{46\!\cdots\!02}{12\!\cdots\!23}a^{25}-\frac{92\!\cdots\!71}{24\!\cdots\!46}a^{24}-\frac{77\!\cdots\!41}{24\!\cdots\!46}a^{23}+\frac{31\!\cdots\!42}{12\!\cdots\!23}a^{22}-\frac{42\!\cdots\!63}{12\!\cdots\!23}a^{21}-\frac{53\!\cdots\!13}{24\!\cdots\!46}a^{20}-\frac{86\!\cdots\!87}{24\!\cdots\!46}a^{19}-\frac{22\!\cdots\!79}{24\!\cdots\!46}a^{18}-\frac{58\!\cdots\!67}{24\!\cdots\!46}a^{17}-\frac{17\!\cdots\!25}{12\!\cdots\!23}a^{16}-\frac{62\!\cdots\!09}{24\!\cdots\!46}a^{15}+\frac{34\!\cdots\!75}{12\!\cdots\!23}a^{14}-\frac{51\!\cdots\!13}{24\!\cdots\!46}a^{13}+\frac{41\!\cdots\!55}{12\!\cdots\!23}a^{12}-\frac{22\!\cdots\!41}{12\!\cdots\!23}a^{11}+\frac{42\!\cdots\!35}{12\!\cdots\!23}a^{10}-\frac{56\!\cdots\!52}{12\!\cdots\!23}a^{9}+\frac{31\!\cdots\!70}{12\!\cdots\!23}a^{8}-\frac{25\!\cdots\!02}{12\!\cdots\!23}a^{7}+\frac{59\!\cdots\!07}{24\!\cdots\!46}a^{6}-\frac{50\!\cdots\!37}{24\!\cdots\!46}a^{5}+\frac{78\!\cdots\!69}{24\!\cdots\!46}a^{4}+\frac{28\!\cdots\!08}{12\!\cdots\!23}a^{3}+\frac{35\!\cdots\!54}{12\!\cdots\!23}a^{2}-\frac{11\!\cdots\!35}{24\!\cdots\!46}a-\frac{53\!\cdots\!45}{12\!\cdots\!23}$ 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

not computed

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:  $43$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^44 - 7*x^43 - 187*x^42 + 1072*x^41 + 17163*x^40 - 70103*x^39 - 997971*x^38 + 2375465*x^37 + 39641205*x^36 - 33307763*x^35 - 1096561800*x^34 - 533702618*x^33 + 20931922170*x^32 + 35314456183*x^31 - 264474855314*x^30 - 793926535475*x^29 + 1942918254504*x^28 + 10265625159992*x^27 - 3458448525180*x^26 - 80753578257977*x^25 - 78276745999632*x^24 + 358613104785724*x^23 + 809785514922531*x^22 - 568771257825877*x^21 - 3562658275456962*x^20 - 2066339006089008*x^19 + 7248452558678380*x^18 + 11554435372453566*x^17 - 3203124978259608*x^16 - 20843198648583718*x^15 - 11861788440019640*x^14 + 14310229227828665*x^13 + 19835277513240846*x^12 + 991200223617035*x^11 - 11320902849209936*x^10 - 5711158023758032*x^9 + 1999942105903247*x^8 + 2439911320594527*x^7 + 307599738065272*x^6 - 336467669357964*x^5 - 115847184197020*x^4 + 5127635333867*x^3 + 5921101695236*x^2 + 200575150497*x - 84271515023)
 
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^44 - 7*x^43 - 187*x^42 + 1072*x^41 + 17163*x^40 - 70103*x^39 - 997971*x^38 + 2375465*x^37 + 39641205*x^36 - 33307763*x^35 - 1096561800*x^34 - 533702618*x^33 + 20931922170*x^32 + 35314456183*x^31 - 264474855314*x^30 - 793926535475*x^29 + 1942918254504*x^28 + 10265625159992*x^27 - 3458448525180*x^26 - 80753578257977*x^25 - 78276745999632*x^24 + 358613104785724*x^23 + 809785514922531*x^22 - 568771257825877*x^21 - 3562658275456962*x^20 - 2066339006089008*x^19 + 7248452558678380*x^18 + 11554435372453566*x^17 - 3203124978259608*x^16 - 20843198648583718*x^15 - 11861788440019640*x^14 + 14310229227828665*x^13 + 19835277513240846*x^12 + 991200223617035*x^11 - 11320902849209936*x^10 - 5711158023758032*x^9 + 1999942105903247*x^8 + 2439911320594527*x^7 + 307599738065272*x^6 - 336467669357964*x^5 - 115847184197020*x^4 + 5127635333867*x^3 + 5921101695236*x^2 + 200575150497*x - 84271515023, 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^44 - 7*x^43 - 187*x^42 + 1072*x^41 + 17163*x^40 - 70103*x^39 - 997971*x^38 + 2375465*x^37 + 39641205*x^36 - 33307763*x^35 - 1096561800*x^34 - 533702618*x^33 + 20931922170*x^32 + 35314456183*x^31 - 264474855314*x^30 - 793926535475*x^29 + 1942918254504*x^28 + 10265625159992*x^27 - 3458448525180*x^26 - 80753578257977*x^25 - 78276745999632*x^24 + 358613104785724*x^23 + 809785514922531*x^22 - 568771257825877*x^21 - 3562658275456962*x^20 - 2066339006089008*x^19 + 7248452558678380*x^18 + 11554435372453566*x^17 - 3203124978259608*x^16 - 20843198648583718*x^15 - 11861788440019640*x^14 + 14310229227828665*x^13 + 19835277513240846*x^12 + 991200223617035*x^11 - 11320902849209936*x^10 - 5711158023758032*x^9 + 1999942105903247*x^8 + 2439911320594527*x^7 + 307599738065272*x^6 - 336467669357964*x^5 - 115847184197020*x^4 + 5127635333867*x^3 + 5921101695236*x^2 + 200575150497*x - 84271515023);
 
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^44 - 7*x^43 - 187*x^42 + 1072*x^41 + 17163*x^40 - 70103*x^39 - 997971*x^38 + 2375465*x^37 + 39641205*x^36 - 33307763*x^35 - 1096561800*x^34 - 533702618*x^33 + 20931922170*x^32 + 35314456183*x^31 - 264474855314*x^30 - 793926535475*x^29 + 1942918254504*x^28 + 10265625159992*x^27 - 3458448525180*x^26 - 80753578257977*x^25 - 78276745999632*x^24 + 358613104785724*x^23 + 809785514922531*x^22 - 568771257825877*x^21 - 3562658275456962*x^20 - 2066339006089008*x^19 + 7248452558678380*x^18 + 11554435372453566*x^17 - 3203124978259608*x^16 - 20843198648583718*x^15 - 11861788440019640*x^14 + 14310229227828665*x^13 + 19835277513240846*x^12 + 991200223617035*x^11 - 11320902849209936*x^10 - 5711158023758032*x^9 + 1999942105903247*x^8 + 2439911320594527*x^7 + 307599738065272*x^6 - 336467669357964*x^5 - 115847184197020*x^4 + 5127635333867*x^3 + 5921101695236*x^2 + 200575150497*x - 84271515023);
 
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

$C_{44}$ (as 44T1):

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)
 
A cyclic group of order 44
The 44 conjugacy class representatives for $C_{44}$
Character table for $C_{44}$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, \(\Q(\zeta_{23})^+\), 22.22.944450376932556277593597370211011550363631641.1

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 $22^{2}$ $44$ $22^{2}$ $44$ $44$ $44$ $44$ $44$ R $44$ ${\href{/padicField/31.11.0.1}{11} }^{4}$ ${\href{/padicField/37.11.0.1}{11} }^{4}$ R $22^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{11}$ $44$ ${\href{/padicField/59.11.0.1}{11} }^{4}$

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
\(23\) Copy content Toggle raw display 23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
\(41\) Copy content Toggle raw display Deg $44$$4$$11$$33$