LMFDB - Number field 40.0.645956895148959968043416875848059904118905976531208178355650775729831936.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 7*x^38 + 40*x^36 - 217*x^34 + 1159*x^32 - 6160*x^30 + 32689*x^28 - 173383*x^26 + 919480*x^24 - 4875913*x^22 + 25856071*x^20 - 43883217*x^18 + 74477880*x^16 - 126396207*x^14 + 214472529*x^12 - 363741840*x^10 + 615940119*x^8 - 1037904273*x^6 + 1721868840*x^4 - 2711943423*x^2 + 3486784401)

gp: K = bnfinit(y^40 - 7*y^38 + 40*y^36 - 217*y^34 + 1159*y^32 - 6160*y^30 + 32689*y^28 - 173383*y^26 + 919480*y^24 - 4875913*y^22 + 25856071*y^20 - 43883217*y^18 + 74477880*y^16 - 126396207*y^14 + 214472529*y^12 - 363741840*y^10 + 615940119*y^8 - 1037904273*y^6 + 1721868840*y^4 - 2711943423*y^2 + 3486784401, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 7*x^38 + 40*x^36 - 217*x^34 + 1159*x^32 - 6160*x^30 + 32689*x^28 - 173383*x^26 + 919480*x^24 - 4875913*x^22 + 25856071*x^20 - 43883217*x^18 + 74477880*x^16 - 126396207*x^14 + 214472529*x^12 - 363741840*x^10 + 615940119*x^8 - 1037904273*x^6 + 1721868840*x^4 - 2711943423*x^2 + 3486784401);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 7*x^38 + 40*x^36 - 217*x^34 + 1159*x^32 - 6160*x^30 + 32689*x^28 - 173383*x^26 + 919480*x^24 - 4875913*x^22 + 25856071*x^20 - 43883217*x^18 + 74477880*x^16 - 126396207*x^14 + 214472529*x^12 - 363741840*x^10 + 615940119*x^8 - 1037904273*x^6 + 1721868840*x^4 - 2711943423*x^2 + 3486784401)

$ x^{40} - 7 x^{38} + 40 x^{36} - 217 x^{34} + 1159 x^{32} - 6160 x^{30} + 32689 x^{28} + \cdots + 3486784401 $ x^40 - 7*x^38 + 40*x^36 - 217*x^34 + 1159*x^32 - 6160*x^30 + 32689*x^28 - 173383*x^26 + 919480*x^24 - 4875913*x^22 + 25856071*x^20 - 43883217*x^18 + 74477880*x^16 - 126396207*x^14 + 214472529*x^12 - 363741840*x^10 + 615940119*x^8 - 1037904273*x^6 + 1721868840*x^4 - 2711943423*x^2 + 3486784401

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:	$645956895148959968043416875848059904118905976531208178355650775729831936$ 645956895148959968043416875848059904118905976531208178355650775729831936 $\medspace = 2^{40}\cdot 11^{36}\cdot 13^{20}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$62.41$	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:	$2\cdot 11^{9/10}13^{1/2}\approx 62.410130178864044$
Ramified primes:	$2$, $11$, $13$ 2, 11, 13	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:	$572=2^{2}\cdot 11\cdot 13$
Dirichlet character group:	$\lbrace$$\chi_{572}(1,·)$, $\chi_{572}(259,·)$, $\chi_{572}(389,·)$, $\chi_{572}(519,·)$, $\chi_{572}(521,·)$, $\chi_{572}(365,·)$, $\chi_{572}(131,·)$, $\chi_{572}(25,·)$, $\chi_{572}(27,·)$, $\chi_{572}(157,·)$, $\chi_{572}(415,·)$, $\chi_{572}(545,·)$, $\chi_{572}(547,·)$, $\chi_{572}(261,·)$, $\chi_{572}(129,·)$, $\chi_{572}(391,·)$, $\chi_{572}(285,·)$, $\chi_{572}(51,·)$, $\chi_{572}(155,·)$, $\chi_{572}(181,·)$, $\chi_{572}(311,·)$, $\chi_{572}(313,·)$, $\chi_{572}(287,·)$, $\chi_{572}(53,·)$, $\chi_{572}(417,·)$, $\chi_{572}(183,·)$, $\chi_{572}(79,·)$, $\chi_{572}(337,·)$, $\chi_{572}(339,·)$, $\chi_{572}(469,·)$, $\chi_{572}(441,·)$, $\chi_{572}(207,·)$, $\chi_{572}(571,·)$, $\chi_{572}(103,·)$, $\chi_{572}(233,·)$, $\chi_{572}(235,·)$, $\chi_{572}(493,·)$, $\chi_{572}(467,·)$, $\chi_{572}(105,·)$, $\chi_{572}(443,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3}a^{21}-\frac{1}{3}a^{19}+\frac{1}{3}a^{17}-\frac{1}{3}a^{15}+\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{232704639}a^{22}-\frac{1}{9}a^{20}-\frac{2}{9}a^{18}-\frac{4}{9}a^{16}+\frac{1}{9}a^{14}+\frac{2}{9}a^{12}+\frac{4}{9}a^{10}-\frac{1}{9}a^{8}-\frac{2}{9}a^{6}-\frac{4}{9}a^{4}+\frac{1}{9}a^{2}-\frac{4875913}{25856071}$, $\frac{1}{698113917}a^{23}-\frac{1}{27}a^{21}-\frac{2}{27}a^{19}-\frac{4}{27}a^{17}-\frac{8}{27}a^{15}+\frac{11}{27}a^{13}-\frac{5}{27}a^{11}-\frac{10}{27}a^{9}+\frac{7}{27}a^{7}-\frac{13}{27}a^{5}+\frac{1}{27}a^{3}-\frac{4875913}{77568213}a$, $\frac{1}{2094341751}a^{24}+\frac{2}{2094341751}a^{22}-\frac{11}{81}a^{20}+\frac{5}{81}a^{18}-\frac{17}{81}a^{16}-\frac{7}{81}a^{14}+\frac{40}{81}a^{12}+\frac{26}{81}a^{10}+\frac{25}{81}a^{8}-\frac{4}{81}a^{6}-\frac{35}{81}a^{4}+\frac{6993386}{77568213}a^{2}-\frac{3956433}{25856071}$, $\frac{1}{6283025253}a^{25}+\frac{2}{6283025253}a^{23}-\frac{11}{243}a^{21}+\frac{5}{243}a^{19}+\frac{64}{243}a^{17}-\frac{7}{243}a^{15}-\frac{41}{243}a^{13}+\frac{107}{243}a^{11}+\frac{106}{243}a^{9}-\frac{4}{243}a^{7}+\frac{46}{243}a^{5}+\frac{84561599}{232704639}a^{3}-\frac{29812504}{77568213}a$, $\frac{1}{18849075759}a^{26}+\frac{2}{18849075759}a^{24}-\frac{1}{819525033}a^{22}-\frac{319}{729}a^{20}+\frac{145}{729}a^{18}-\frac{331}{729}a^{16}+\frac{283}{729}a^{14}+\frac{269}{729}a^{12}-\frac{56}{729}a^{10}+\frac{158}{729}a^{8}+\frac{127}{729}a^{6}+\frac{6993386}{698113917}a^{4}-\frac{1318811}{77568213}a^{2}+\frac{32439}{1124177}$, $\frac{1}{56547227277}a^{27}+\frac{2}{56547227277}a^{25}-\frac{1}{2458575099}a^{23}-\frac{319}{2187}a^{21}-\frac{584}{2187}a^{19}+\frac{398}{2187}a^{17}+\frac{283}{2187}a^{15}+\frac{998}{2187}a^{13}-\frac{785}{2187}a^{11}+\frac{887}{2187}a^{9}+\frac{856}{2187}a^{7}+\frac{6993386}{2094341751}a^{5}-\frac{1318811}{232704639}a^{3}+\frac{10813}{1124177}a$, $\frac{1}{169641681831}a^{28}+\frac{2}{169641681831}a^{26}-\frac{1}{7375725297}a^{24}+\frac{143}{169641681831}a^{22}-\frac{584}{6561}a^{20}+\frac{398}{6561}a^{18}+\frac{2470}{6561}a^{16}-\frac{1189}{6561}a^{14}-\frac{785}{6561}a^{12}+\frac{3074}{6561}a^{10}-\frac{1331}{6561}a^{8}+\frac{6993386}{6283025253}a^{6}-\frac{1318811}{698113917}a^{4}+\frac{10813}{3372531}a^{2}-\frac{140694}{25856071}$, $\frac{1}{508925045493}a^{29}+\frac{2}{508925045493}a^{27}-\frac{1}{22127175891}a^{25}+\frac{143}{508925045493}a^{23}-\frac{584}{19683}a^{21}+\frac{6959}{19683}a^{19}-\frac{4091}{19683}a^{17}+\frac{5372}{19683}a^{15}-\frac{785}{19683}a^{13}-\frac{3487}{19683}a^{11}-\frac{7892}{19683}a^{9}-\frac{6276031867}{18849075759}a^{7}+\frac{696795106}{2094341751}a^{5}-\frac{3361718}{10117593}a^{3}+\frac{25715377}{77568213}a$, $\frac{1}{1526775136479}a^{30}+\frac{2}{1526775136479}a^{28}-\frac{1}{66381527673}a^{26}+\frac{143}{1526775136479}a^{24}-\frac{794}{1526775136479}a^{22}-\frac{25846}{59049}a^{20}+\frac{9031}{59049}a^{18}-\frac{7750}{59049}a^{16}-\frac{27029}{59049}a^{14}+\frac{22757}{59049}a^{12}+\frac{24913}{59049}a^{10}+\frac{6993386}{56547227277}a^{8}-\frac{1318811}{6283025253}a^{6}+\frac{10813}{30352779}a^{4}-\frac{46898}{77568213}a^{2}+\frac{26529}{25856071}$, $\frac{1}{4580325409437}a^{31}+\frac{2}{4580325409437}a^{29}-\frac{1}{199144583019}a^{27}+\frac{143}{4580325409437}a^{25}-\frac{794}{4580325409437}a^{23}-\frac{25846}{177147}a^{21}+\frac{68080}{177147}a^{19}-\frac{66799}{177147}a^{17}+\frac{32020}{177147}a^{15}+\frac{22757}{177147}a^{13}+\frac{83962}{177147}a^{11}+\frac{56554220663}{169641681831}a^{9}-\frac{6284344064}{18849075759}a^{7}+\frac{30363592}{91058337}a^{5}-\frac{77615111}{232704639}a^{3}+\frac{25882600}{77568213}a$, $\frac{1}{13740976228311}a^{32}+\frac{2}{13740976228311}a^{30}-\frac{1}{597433749057}a^{28}+\frac{143}{13740976228311}a^{26}-\frac{794}{13740976228311}a^{24}+\frac{4271}{13740976228311}a^{22}-\frac{227165}{531441}a^{20}+\frac{228446}{531441}a^{18}-\frac{86078}{531441}a^{16}+\frac{140855}{531441}a^{14}-\frac{211283}{531441}a^{12}+\frac{6993386}{508925045493}a^{10}-\frac{1318811}{56547227277}a^{8}+\frac{10813}{273175011}a^{6}-\frac{46898}{698113917}a^{4}+\frac{8843}{77568213}a^{2}-\frac{5001}{25856071}$, $\frac{1}{41222928684933}a^{33}+\frac{2}{41222928684933}a^{31}-\frac{1}{1792301247171}a^{29}+\frac{143}{41222928684933}a^{27}-\frac{794}{41222928684933}a^{25}+\frac{4271}{41222928684933}a^{23}-\frac{227165}{1594323}a^{21}+\frac{759887}{1594323}a^{19}-\frac{86078}{1594323}a^{17}+\frac{140855}{1594323}a^{15}-\frac{211283}{1594323}a^{13}+\frac{6993386}{1526775136479}a^{11}-\frac{1318811}{169641681831}a^{9}+\frac{10813}{819525033}a^{7}-\frac{46898}{2094341751}a^{5}+\frac{8843}{232704639}a^{3}-\frac{1667}{25856071}a$, $\frac{1}{123668786054799}a^{34}+\frac{2}{123668786054799}a^{32}-\frac{1}{5376903741513}a^{30}+\frac{143}{123668786054799}a^{28}-\frac{794}{123668786054799}a^{26}+\frac{4271}{123668786054799}a^{24}-\frac{22751}{123668786054799}a^{22}-\frac{834436}{4782969}a^{20}-\frac{1680401}{4782969}a^{18}+\frac{140855}{4782969}a^{16}-\frac{211283}{4782969}a^{14}+\frac{6993386}{4580325409437}a^{12}-\frac{1318811}{508925045493}a^{10}+\frac{10813}{2458575099}a^{8}-\frac{46898}{6283025253}a^{6}+\frac{8843}{698113917}a^{4}-\frac{1667}{77568213}a^{2}+\frac{942}{25856071}$, $\frac{1}{371006358164397}a^{35}+\frac{2}{371006358164397}a^{33}-\frac{1}{16130711224539}a^{31}+\frac{143}{371006358164397}a^{29}-\frac{794}{371006358164397}a^{27}+\frac{4271}{371006358164397}a^{25}-\frac{22751}{371006358164397}a^{23}-\frac{834436}{14348907}a^{21}-\frac{6463370}{14348907}a^{19}-\frac{4642114}{14348907}a^{17}+\frac{4571686}{14348907}a^{15}-\frac{4580318416051}{13740976228311}a^{13}+\frac{508923726682}{1526775136479}a^{11}-\frac{2458564286}{7375725297}a^{9}+\frac{6282978355}{18849075759}a^{7}-\frac{698105074}{2094341751}a^{5}+\frac{77566546}{232704639}a^{3}-\frac{25855129}{77568213}a$, $\frac{1}{11\!\cdots\!91}a^{36}+\frac{2}{11\!\cdots\!91}a^{34}-\frac{1}{48392133673617}a^{32}+\frac{143}{11\!\cdots\!91}a^{30}-\frac{794}{11\!\cdots\!91}a^{28}+\frac{4271}{11\!\cdots\!91}a^{26}-\frac{22751}{11\!\cdots\!91}a^{24}+\frac{120818}{11\!\cdots\!91}a^{22}+\frac{17451475}{43046721}a^{20}+\frac{14489762}{43046721}a^{18}-\frac{211283}{43046721}a^{16}+\frac{6993386}{41222928684933}a^{14}-\frac{1318811}{4580325409437}a^{12}+\frac{10813}{22127175891}a^{10}-\frac{46898}{56547227277}a^{8}+\frac{8843}{6283025253}a^{6}-\frac{1667}{698113917}a^{4}+\frac{314}{77568213}a^{2}-\frac{177}{25856071}$, $\frac{1}{33\!\cdots\!73}a^{37}+\frac{2}{33\!\cdots\!73}a^{35}-\frac{1}{145176401020851}a^{33}+\frac{143}{33\!\cdots\!73}a^{31}-\frac{794}{33\!\cdots\!73}a^{29}+\frac{4271}{33\!\cdots\!73}a^{27}-\frac{22751}{33\!\cdots\!73}a^{25}+\frac{120818}{33\!\cdots\!73}a^{23}+\frac{17451475}{129140163}a^{21}-\frac{28556959}{129140163}a^{19}+\frac{42835438}{129140163}a^{17}+\frac{6993386}{123668786054799}a^{15}-\frac{1318811}{13740976228311}a^{13}+\frac{10813}{66381527673}a^{11}-\frac{46898}{169641681831}a^{9}+\frac{8843}{18849075759}a^{7}-\frac{1667}{2094341751}a^{5}+\frac{314}{232704639}a^{3}-\frac{59}{25856071}a$, $\frac{1}{10\!\cdots\!19}a^{38}+\frac{2}{10\!\cdots\!19}a^{36}-\frac{1}{435529203062553}a^{34}+\frac{143}{10\!\cdots\!19}a^{32}-\frac{794}{10\!\cdots\!19}a^{30}+\frac{4271}{10\!\cdots\!19}a^{28}-\frac{22751}{10\!\cdots\!19}a^{26}+\frac{120818}{10\!\cdots\!19}a^{24}-\frac{640967}{10\!\cdots\!19}a^{22}-\frac{28556959}{387420489}a^{20}+\frac{42835438}{387420489}a^{18}+\frac{6993386}{371006358164397}a^{16}-\frac{1318811}{41222928684933}a^{14}+\frac{10813}{199144583019}a^{12}-\frac{46898}{508925045493}a^{10}+\frac{8843}{56547227277}a^{8}-\frac{1667}{6283025253}a^{6}+\frac{314}{698113917}a^{4}-\frac{59}{77568213}a^{2}+\frac{33}{25856071}$, $\frac{1}{30\!\cdots\!57}a^{39}+\frac{2}{30\!\cdots\!57}a^{37}-\frac{1}{13\!\cdots\!59}a^{35}+\frac{143}{30\!\cdots\!57}a^{33}-\frac{794}{30\!\cdots\!57}a^{31}+\frac{4271}{30\!\cdots\!57}a^{29}-\frac{22751}{30\!\cdots\!57}a^{27}+\frac{120818}{30\!\cdots\!57}a^{25}-\frac{640967}{30\!\cdots\!57}a^{23}-\frac{28556959}{1162261467}a^{21}+\frac{42835438}{1162261467}a^{19}+\frac{6993386}{11\!\cdots\!91}a^{17}-\frac{1318811}{123668786054799}a^{15}+\frac{10813}{597433749057}a^{13}-\frac{46898}{1526775136479}a^{11}+\frac{8843}{169641681831}a^{9}-\frac{1667}{18849075759}a^{7}+\frac{314}{2094341751}a^{5}-\frac{59}{232704639}a^{3}+\frac{11}{25856071}a$ 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, 1/3*a^21 - 1/3*a^19 + 1/3*a^17 - 1/3*a^15 + 1/3*a^13 - 1/3*a^11 + 1/3*a^9 - 1/3*a^7 + 1/3*a^5 - 1/3*a^3 + 1/3*a, 1/232704639*a^22 - 1/9*a^20 - 2/9*a^18 - 4/9*a^16 + 1/9*a^14 + 2/9*a^12 + 4/9*a^10 - 1/9*a^8 - 2/9*a^6 - 4/9*a^4 + 1/9*a^2 - 4875913/25856071, 1/698113917*a^23 - 1/27*a^21 - 2/27*a^19 - 4/27*a^17 - 8/27*a^15 + 11/27*a^13 - 5/27*a^11 - 10/27*a^9 + 7/27*a^7 - 13/27*a^5 + 1/27*a^3 - 4875913/77568213*a, 1/2094341751*a^24 + 2/2094341751*a^22 - 11/81*a^20 + 5/81*a^18 - 17/81*a^16 - 7/81*a^14 + 40/81*a^12 + 26/81*a^10 + 25/81*a^8 - 4/81*a^6 - 35/81*a^4 + 6993386/77568213*a^2 - 3956433/25856071, 1/6283025253*a^25 + 2/6283025253*a^23 - 11/243*a^21 + 5/243*a^19 + 64/243*a^17 - 7/243*a^15 - 41/243*a^13 + 107/243*a^11 + 106/243*a^9 - 4/243*a^7 + 46/243*a^5 + 84561599/232704639*a^3 - 29812504/77568213*a, 1/18849075759*a^26 + 2/18849075759*a^24 - 1/819525033*a^22 - 319/729*a^20 + 145/729*a^18 - 331/729*a^16 + 283/729*a^14 + 269/729*a^12 - 56/729*a^10 + 158/729*a^8 + 127/729*a^6 + 6993386/698113917*a^4 - 1318811/77568213*a^2 + 32439/1124177, 1/56547227277*a^27 + 2/56547227277*a^25 - 1/2458575099*a^23 - 319/2187*a^21 - 584/2187*a^19 + 398/2187*a^17 + 283/2187*a^15 + 998/2187*a^13 - 785/2187*a^11 + 887/2187*a^9 + 856/2187*a^7 + 6993386/2094341751*a^5 - 1318811/232704639*a^3 + 10813/1124177*a, 1/169641681831*a^28 + 2/169641681831*a^26 - 1/7375725297*a^24 + 143/169641681831*a^22 - 584/6561*a^20 + 398/6561*a^18 + 2470/6561*a^16 - 1189/6561*a^14 - 785/6561*a^12 + 3074/6561*a^10 - 1331/6561*a^8 + 6993386/6283025253*a^6 - 1318811/698113917*a^4 + 10813/3372531*a^2 - 140694/25856071, 1/508925045493*a^29 + 2/508925045493*a^27 - 1/22127175891*a^25 + 143/508925045493*a^23 - 584/19683*a^21 + 6959/19683*a^19 - 4091/19683*a^17 + 5372/19683*a^15 - 785/19683*a^13 - 3487/19683*a^11 - 7892/19683*a^9 - 6276031867/18849075759*a^7 + 696795106/2094341751*a^5 - 3361718/10117593*a^3 + 25715377/77568213*a, 1/1526775136479*a^30 + 2/1526775136479*a^28 - 1/66381527673*a^26 + 143/1526775136479*a^24 - 794/1526775136479*a^22 - 25846/59049*a^20 + 9031/59049*a^18 - 7750/59049*a^16 - 27029/59049*a^14 + 22757/59049*a^12 + 24913/59049*a^10 + 6993386/56547227277*a^8 - 1318811/6283025253*a^6 + 10813/30352779*a^4 - 46898/77568213*a^2 + 26529/25856071, 1/4580325409437*a^31 + 2/4580325409437*a^29 - 1/199144583019*a^27 + 143/4580325409437*a^25 - 794/4580325409437*a^23 - 25846/177147*a^21 + 68080/177147*a^19 - 66799/177147*a^17 + 32020/177147*a^15 + 22757/177147*a^13 + 83962/177147*a^11 + 56554220663/169641681831*a^9 - 6284344064/18849075759*a^7 + 30363592/91058337*a^5 - 77615111/232704639*a^3 + 25882600/77568213*a, 1/13740976228311*a^32 + 2/13740976228311*a^30 - 1/597433749057*a^28 + 143/13740976228311*a^26 - 794/13740976228311*a^24 + 4271/13740976228311*a^22 - 227165/531441*a^20 + 228446/531441*a^18 - 86078/531441*a^16 + 140855/531441*a^14 - 211283/531441*a^12 + 6993386/508925045493*a^10 - 1318811/56547227277*a^8 + 10813/273175011*a^6 - 46898/698113917*a^4 + 8843/77568213*a^2 - 5001/25856071, 1/41222928684933*a^33 + 2/41222928684933*a^31 - 1/1792301247171*a^29 + 143/41222928684933*a^27 - 794/41222928684933*a^25 + 4271/41222928684933*a^23 - 227165/1594323*a^21 + 759887/1594323*a^19 - 86078/1594323*a^17 + 140855/1594323*a^15 - 211283/1594323*a^13 + 6993386/1526775136479*a^11 - 1318811/169641681831*a^9 + 10813/819525033*a^7 - 46898/2094341751*a^5 + 8843/232704639*a^3 - 1667/25856071*a, 1/123668786054799*a^34 + 2/123668786054799*a^32 - 1/5376903741513*a^30 + 143/123668786054799*a^28 - 794/123668786054799*a^26 + 4271/123668786054799*a^24 - 22751/123668786054799*a^22 - 834436/4782969*a^20 - 1680401/4782969*a^18 + 140855/4782969*a^16 - 211283/4782969*a^14 + 6993386/4580325409437*a^12 - 1318811/508925045493*a^10 + 10813/2458575099*a^8 - 46898/6283025253*a^6 + 8843/698113917*a^4 - 1667/77568213*a^2 + 942/25856071, 1/371006358164397*a^35 + 2/371006358164397*a^33 - 1/16130711224539*a^31 + 143/371006358164397*a^29 - 794/371006358164397*a^27 + 4271/371006358164397*a^25 - 22751/371006358164397*a^23 - 834436/14348907*a^21 - 6463370/14348907*a^19 - 4642114/14348907*a^17 + 4571686/14348907*a^15 - 4580318416051/13740976228311*a^13 + 508923726682/1526775136479*a^11 - 2458564286/7375725297*a^9 + 6282978355/18849075759*a^7 - 698105074/2094341751*a^5 + 77566546/232704639*a^3 - 25855129/77568213*a, 1/1113019074493191*a^36 + 2/1113019074493191*a^34 - 1/48392133673617*a^32 + 143/1113019074493191*a^30 - 794/1113019074493191*a^28 + 4271/1113019074493191*a^26 - 22751/1113019074493191*a^24 + 120818/1113019074493191*a^22 + 17451475/43046721*a^20 + 14489762/43046721*a^18 - 211283/43046721*a^16 + 6993386/41222928684933*a^14 - 1318811/4580325409437*a^12 + 10813/22127175891*a^10 - 46898/56547227277*a^8 + 8843/6283025253*a^6 - 1667/698113917*a^4 + 314/77568213*a^2 - 177/25856071, 1/3339057223479573*a^37 + 2/3339057223479573*a^35 - 1/145176401020851*a^33 + 143/3339057223479573*a^31 - 794/3339057223479573*a^29 + 4271/3339057223479573*a^27 - 22751/3339057223479573*a^25 + 120818/3339057223479573*a^23 + 17451475/129140163*a^21 - 28556959/129140163*a^19 + 42835438/129140163*a^17 + 6993386/123668786054799*a^15 - 1318811/13740976228311*a^13 + 10813/66381527673*a^11 - 46898/169641681831*a^9 + 8843/18849075759*a^7 - 1667/2094341751*a^5 + 314/232704639*a^3 - 59/25856071*a, 1/10017171670438719*a^38 + 2/10017171670438719*a^36 - 1/435529203062553*a^34 + 143/10017171670438719*a^32 - 794/10017171670438719*a^30 + 4271/10017171670438719*a^28 - 22751/10017171670438719*a^26 + 120818/10017171670438719*a^24 - 640967/10017171670438719*a^22 - 28556959/387420489*a^20 + 42835438/387420489*a^18 + 6993386/371006358164397*a^16 - 1318811/41222928684933*a^14 + 10813/199144583019*a^12 - 46898/508925045493*a^10 + 8843/56547227277*a^8 - 1667/6283025253*a^6 + 314/698113917*a^4 - 59/77568213*a^2 + 33/25856071, 1/30051515011316157*a^39 + 2/30051515011316157*a^37 - 1/1306587609187659*a^35 + 143/30051515011316157*a^33 - 794/30051515011316157*a^31 + 4271/30051515011316157*a^29 - 22751/30051515011316157*a^27 + 120818/30051515011316157*a^25 - 640967/30051515011316157*a^23 - 28556959/1162261467*a^21 + 42835438/1162261467*a^19 + 6993386/1113019074493191*a^17 - 1318811/123668786054799*a^15 + 10813/597433749057*a^13 - 46898/1526775136479*a^11 + 8843/169641681831*a^9 - 1667/18849075759*a^7 + 314/2094341751*a^5 - 59/232704639*a^3 + 11/25856071*a

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:	$19$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -\frac{75316}{371006358164397} a^{37} - \frac{7020022316929}{371006358164397} a^{15} $ -(75316)/(371006358164397)a^(37) - (7020022316929)/(371006358164397)a^(15) (order $44$)	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^40 - 7*x^38 + 40*x^36 - 217*x^34 + 1159*x^32 - 6160*x^30 + 32689*x^28 - 173383*x^26 + 919480*x^24 - 4875913*x^22 + 25856071*x^20 - 43883217*x^18 + 74477880*x^16 - 126396207*x^14 + 214472529*x^12 - 363741840*x^10 + 615940119*x^8 - 1037904273*x^6 + 1721868840*x^4 - 2711943423*x^2 + 3486784401)

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 - 7*x^38 + 40*x^36 - 217*x^34 + 1159*x^32 - 6160*x^30 + 32689*x^28 - 173383*x^26 + 919480*x^24 - 4875913*x^22 + 25856071*x^20 - 43883217*x^18 + 74477880*x^16 - 126396207*x^14 + 214472529*x^12 - 363741840*x^10 + 615940119*x^8 - 1037904273*x^6 + 1721868840*x^4 - 2711943423*x^2 + 3486784401, 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 - 7*x^38 + 40*x^36 - 217*x^34 + 1159*x^32 - 6160*x^30 + 32689*x^28 - 173383*x^26 + 919480*x^24 - 4875913*x^22 + 25856071*x^20 - 43883217*x^18 + 74477880*x^16 - 126396207*x^14 + 214472529*x^12 - 363741840*x^10 + 615940119*x^8 - 1037904273*x^6 + 1721868840*x^4 - 2711943423*x^2 + 3486784401);

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 - 7*x^38 + 40*x^36 - 217*x^34 + 1159*x^32 - 6160*x^30 + 32689*x^28 - 173383*x^26 + 919480*x^24 - 4875913*x^22 + 25856071*x^20 - 43883217*x^18 + 74477880*x^16 - 126396207*x^14 + 214472529*x^12 - 363741840*x^10 + 615940119*x^8 - 1037904273*x^6 + 1721868840*x^4 - 2711943423*x^2 + 3486784401);

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{-1}) $, $\Q(\sqrt{-13}) $, $\Q(\sqrt{13}) $, $\Q(\sqrt{-143}) $, $\Q(\sqrt{143}) $, $\Q(\sqrt{11}) $, $\Q(\sqrt{-11}) $, $\Q(i, \sqrt{13})$, $\Q(i, \sqrt{143})$, $\Q(i, \sqrt{11})$, $\Q(\sqrt{11}, \sqrt{-13})$, $\Q(\sqrt{-11}, \sqrt{-13})$, $\Q(\sqrt{-11}, \sqrt{13})$, $\Q(\sqrt{11}, \sqrt{13})$, $\Q(\zeta_{11})^+$, 8.0.107049369856.1, 10.0.219503494144.1, 10.0.81500110851208192.3, 10.10.79589952003133.1, 10.0.875489472034463.1, 10.10.896501219363290112.1, $\Q(\zeta_{44})^+$, $\Q(\zeta_{11})$, 20.0.6642268068759223286357626127908864.1, 20.0.803714436319866017649272761476972544.4, $\Q(\zeta_{44})$, 20.0.803714436319866017649272761476972544.2, 20.0.803714436319866017649272761476972544.5, 20.0.766481815643182771348259698369.1, 20.20.803714436319866017649272761476972544.1

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	${\href{/padicField/3.10.0.1}{10} }^{4}$	${\href{/padicField/5.10.0.1}{10} }^{4}$	${\href{/padicField/7.10.0.1}{10} }^{4}$	R	R	${\href{/padicField/17.10.0.1}{10} }^{4}$	${\href{/padicField/19.10.0.1}{10} }^{4}$	${\href{/padicField/23.2.0.1}{2} }^{20}$	${\href{/padicField/29.10.0.1}{10} }^{4}$	${\href{/padicField/31.10.0.1}{10} }^{4}$	${\href{/padicField/37.10.0.1}{10} }^{4}$	${\href{/padicField/41.10.0.1}{10} }^{4}$	${\href{/padicField/43.2.0.1}{2} }^{20}$	${\href{/padicField/47.10.0.1}{10} }^{4}$	${\href{/padicField/53.5.0.1}{5} }^{8}$	${\href{/padicField/59.10.0.1}{10} }^{4}$

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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2		Deg $20$	$2$	$10$	$20$
$2$ 2		Deg $20$	$2$	$10$	$20$
$11$ 11	11.20.18.1	$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$	$10$	$2$	$18$	20T3	$[\ ]_{10}^{2}$
$11$ 11	11.20.18.1	$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$	$10$	$2$	$18$	20T3	$[\ ]_{10}^{2}$
$13$ 13	13.20.10.1	$x^{20} + 1300 x^{19} + 760630 x^{18} + 263792100 x^{17} + 60057199605 x^{16} + 9380582749214 x^{15} + 1018311780763300 x^{14} + 75907334238787106 x^{13} + 3723649138838156452 x^{12} + 108997457660124507008 x^{11} + 1475055936539742904023 x^{10} + 1416974086834508372240 x^{9} + 629829636823684464902 x^{8} + 192953738203144224036 x^{7} + 798096976259371806456 x^{6} + 10514315760388324731550 x^{5} + 12773049284749524150248 x^{4} + 15044084853570106847092 x^{3} + 5322693419045842540420 x^{2} + 2225999864943544940488 x + 2898376971411614164801$	$2$	$10$	$10$	20T3	$[\ ]_{2}^{10}$
$13$ 13	13.20.10.1	$x^{20} + 1300 x^{19} + 760630 x^{18} + 263792100 x^{17} + 60057199605 x^{16} + 9380582749214 x^{15} + 1018311780763300 x^{14} + 75907334238787106 x^{13} + 3723649138838156452 x^{12} + 108997457660124507008 x^{11} + 1475055936539742904023 x^{10} + 1416974086834508372240 x^{9} + 629829636823684464902 x^{8} + 192953738203144224036 x^{7} + 798096976259371806456 x^{6} + 10514315760388324731550 x^{5} + 12773049284749524150248 x^{4} + 15044084853570106847092 x^{3} + 5322693419045842540420 x^{2} + 2225999864943544940488 x + 2898376971411614164801$	$2$	$10$	$10$	20T3	$[\ ]_{2}^{10}$

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