LMFDB - Number field 40.0.2712047505327472012144594086234943329183650768987920496114727387136.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 3*x^38 + 5*x^36 + 3*x^34 - 11*x^32 - 45*x^30 - 91*x^28 - 93*x^26 + 85*x^24 + 627*x^22 + 1541*x^20 + 2508*x^18 + 1360*x^16 - 5952*x^14 - 23296*x^12 - 46080*x^10 - 45056*x^8 + 49152*x^6 + 327680*x^4 + 786432*x^2 + 1048576)

gp: K = bnfinit(y^40 + 3*y^38 + 5*y^36 + 3*y^34 - 11*y^32 - 45*y^30 - 91*y^28 - 93*y^26 + 85*y^24 + 627*y^22 + 1541*y^20 + 2508*y^18 + 1360*y^16 - 5952*y^14 - 23296*y^12 - 46080*y^10 - 45056*y^8 + 49152*y^6 + 327680*y^4 + 786432*y^2 + 1048576, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 + 3*x^38 + 5*x^36 + 3*x^34 - 11*x^32 - 45*x^30 - 91*x^28 - 93*x^26 + 85*x^24 + 627*x^22 + 1541*x^20 + 2508*x^18 + 1360*x^16 - 5952*x^14 - 23296*x^12 - 46080*x^10 - 45056*x^8 + 49152*x^6 + 327680*x^4 + 786432*x^2 + 1048576);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 + 3*x^38 + 5*x^36 + 3*x^34 - 11*x^32 - 45*x^30 - 91*x^28 - 93*x^26 + 85*x^24 + 627*x^22 + 1541*x^20 + 2508*x^18 + 1360*x^16 - 5952*x^14 - 23296*x^12 - 46080*x^10 - 45056*x^8 + 49152*x^6 + 327680*x^4 + 786432*x^2 + 1048576)

$ x^{40} + 3 x^{38} + 5 x^{36} + 3 x^{34} - 11 x^{32} - 45 x^{30} - 91 x^{28} - 93 x^{26} + 85 x^{24} + \cdots + 1048576 $ x^40 + 3*x^38 + 5*x^36 + 3*x^34 - 11*x^32 - 45*x^30 - 91*x^28 - 93*x^26 + 85*x^24 + 627*x^22 + 1541*x^20 + 2508*x^18 + 1360*x^16 - 5952*x^14 - 23296*x^12 - 46080*x^10 - 45056*x^8 + 49152*x^6 + 327680*x^4 + 786432*x^2 + 1048576

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:	$2712047505327472012144594086234943329183650768987920496114727387136$ 2712047505327472012144594086234943329183650768987920496114727387136 $\medspace = 2^{40}\cdot 7^{20}\cdot 11^{36}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$45.80$	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:	$2\cdot 7^{1/2}11^{9/10}\approx 45.79651518704091$
Ramified primes:	$2$, $7$, $11$ 2, 7, 11	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:	$308=2^{2}\cdot 7\cdot 11$
Dirichlet character group:	$\lbrace$$\chi_{308}(1,·)$, $\chi_{308}(265,·)$, $\chi_{308}(139,·)$, $\chi_{308}(13,·)$, $\chi_{308}(15,·)$, $\chi_{308}(279,·)$, $\chi_{308}(153,·)$, $\chi_{308}(27,·)$, $\chi_{308}(29,·)$, $\chi_{308}(69,·)$, $\chi_{308}(155,·)$, $\chi_{308}(293,·)$, $\chi_{308}(295,·)$, $\chi_{308}(41,·)$, $\chi_{308}(43,·)$, $\chi_{308}(307,·)$, $\chi_{308}(181,·)$, $\chi_{308}(183,·)$, $\chi_{308}(111,·)$, $\chi_{308}(57,·)$, $\chi_{308}(195,·)$, $\chi_{308}(197,·)$, $\chi_{308}(71,·)$, $\chi_{308}(141,·)$, $\chi_{308}(83,·)$, $\chi_{308}(85,·)$, $\chi_{308}(267,·)$, $\chi_{308}(223,·)$, $\chi_{308}(225,·)$, $\chi_{308}(97,·)$, $\chi_{308}(167,·)$, $\chi_{308}(237,·)$, $\chi_{308}(239,·)$, $\chi_{308}(113,·)$, $\chi_{308}(211,·)$, $\chi_{308}(169,·)$, $\chi_{308}(251,·)$, $\chi_{308}(281,·)$, $\chi_{308}(125,·)$, $\chi_{308}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{6164}a^{22}-\frac{1}{4}a^{20}+\frac{1}{4}a^{18}-\frac{1}{4}a^{16}+\frac{1}{4}a^{14}-\frac{1}{4}a^{12}+\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}+\frac{627}{1541}$, $\frac{1}{12328}a^{23}-\frac{1}{8}a^{21}+\frac{1}{8}a^{19}-\frac{1}{8}a^{17}+\frac{1}{8}a^{15}-\frac{1}{8}a^{13}+\frac{1}{8}a^{11}-\frac{1}{8}a^{9}+\frac{1}{8}a^{7}-\frac{1}{8}a^{5}+\frac{1}{8}a^{3}+\frac{627}{3082}a$, $\frac{1}{24656}a^{24}-\frac{1}{24656}a^{22}+\frac{5}{16}a^{20}+\frac{3}{16}a^{18}+\frac{5}{16}a^{16}+\frac{3}{16}a^{14}+\frac{5}{16}a^{12}+\frac{3}{16}a^{10}+\frac{5}{16}a^{8}+\frac{3}{16}a^{6}+\frac{5}{16}a^{4}-\frac{457}{3082}a^{2}-\frac{542}{1541}$, $\frac{1}{49312}a^{25}-\frac{1}{49312}a^{23}+\frac{5}{32}a^{21}+\frac{3}{32}a^{19}-\frac{11}{32}a^{17}-\frac{13}{32}a^{15}+\frac{5}{32}a^{13}+\frac{3}{32}a^{11}-\frac{11}{32}a^{9}-\frac{13}{32}a^{7}+\frac{5}{32}a^{5}-\frac{457}{6164}a^{3}-\frac{271}{1541}a$, $\frac{1}{98624}a^{26}-\frac{1}{98624}a^{24}-\frac{7}{98624}a^{22}-\frac{29}{64}a^{20}+\frac{21}{64}a^{18}-\frac{13}{64}a^{16}+\frac{5}{64}a^{14}+\frac{3}{64}a^{12}-\frac{11}{64}a^{10}+\frac{19}{64}a^{8}-\frac{27}{64}a^{6}-\frac{457}{12328}a^{4}-\frac{271}{3082}a^{2}-\frac{178}{1541}$, $\frac{1}{197248}a^{27}-\frac{1}{197248}a^{25}-\frac{7}{197248}a^{23}-\frac{29}{128}a^{21}+\frac{21}{128}a^{19}+\frac{51}{128}a^{17}-\frac{59}{128}a^{15}+\frac{3}{128}a^{13}-\frac{11}{128}a^{11}-\frac{45}{128}a^{9}+\frac{37}{128}a^{7}-\frac{457}{24656}a^{5}-\frac{271}{6164}a^{3}-\frac{89}{1541}a$, $\frac{1}{394496}a^{28}-\frac{1}{394496}a^{26}-\frac{7}{394496}a^{24}-\frac{17}{394496}a^{22}-\frac{107}{256}a^{20}+\frac{51}{256}a^{18}+\frac{69}{256}a^{16}+\frac{3}{256}a^{14}-\frac{11}{256}a^{12}-\frac{45}{256}a^{10}-\frac{91}{256}a^{8}-\frac{457}{49312}a^{6}-\frac{271}{12328}a^{4}-\frac{89}{3082}a^{2}+\frac{2}{1541}$, $\frac{1}{788992}a^{29}-\frac{1}{788992}a^{27}-\frac{7}{788992}a^{25}-\frac{17}{788992}a^{23}-\frac{107}{512}a^{21}-\frac{205}{512}a^{19}-\frac{187}{512}a^{17}-\frac{253}{512}a^{15}-\frac{11}{512}a^{13}-\frac{45}{512}a^{11}-\frac{91}{512}a^{9}-\frac{457}{98624}a^{7}-\frac{271}{24656}a^{5}-\frac{89}{6164}a^{3}+\frac{1}{1541}a$, $\frac{1}{1577984}a^{30}-\frac{1}{1577984}a^{28}-\frac{7}{1577984}a^{26}-\frac{17}{1577984}a^{24}-\frac{1}{68608}a^{22}+\frac{307}{1024}a^{20}+\frac{325}{1024}a^{18}-\frac{253}{1024}a^{16}-\frac{11}{1024}a^{14}-\frac{45}{1024}a^{12}-\frac{91}{1024}a^{10}-\frac{457}{197248}a^{8}-\frac{271}{49312}a^{6}-\frac{89}{12328}a^{4}+\frac{1}{3082}a^{2}+\frac{2}{67}$, $\frac{1}{3155968}a^{31}-\frac{1}{3155968}a^{29}-\frac{7}{3155968}a^{27}-\frac{17}{3155968}a^{25}-\frac{1}{137216}a^{23}+\frac{307}{2048}a^{21}-\frac{699}{2048}a^{19}+\frac{771}{2048}a^{17}+\frac{1013}{2048}a^{15}-\frac{45}{2048}a^{13}-\frac{91}{2048}a^{11}-\frac{457}{394496}a^{9}-\frac{271}{98624}a^{7}-\frac{89}{24656}a^{5}+\frac{1}{6164}a^{3}+\frac{1}{67}a$, $\frac{1}{6311936}a^{32}-\frac{1}{6311936}a^{30}-\frac{7}{6311936}a^{28}-\frac{17}{6311936}a^{26}-\frac{1}{274432}a^{24}-\frac{1}{6311936}a^{22}-\frac{699}{4096}a^{20}+\frac{771}{4096}a^{18}+\frac{1013}{4096}a^{16}-\frac{45}{4096}a^{14}-\frac{91}{4096}a^{12}-\frac{457}{788992}a^{10}-\frac{271}{197248}a^{8}-\frac{89}{49312}a^{6}+\frac{1}{12328}a^{4}+\frac{1}{134}a^{2}+\frac{34}{1541}$, $\frac{1}{12623872}a^{33}-\frac{1}{12623872}a^{31}-\frac{7}{12623872}a^{29}-\frac{17}{12623872}a^{27}-\frac{1}{548864}a^{25}-\frac{1}{12623872}a^{23}-\frac{699}{8192}a^{21}-\frac{3325}{8192}a^{19}+\frac{1013}{8192}a^{17}-\frac{45}{8192}a^{15}+\frac{4005}{8192}a^{13}+\frac{788535}{1577984}a^{11}+\frac{196977}{394496}a^{9}+\frac{49223}{98624}a^{7}-\frac{12327}{24656}a^{5}-\frac{133}{268}a^{3}-\frac{1507}{3082}a$, $\frac{1}{25247744}a^{34}-\frac{1}{25247744}a^{32}-\frac{7}{25247744}a^{30}-\frac{17}{25247744}a^{28}-\frac{1}{1097728}a^{26}-\frac{1}{25247744}a^{24}+\frac{89}{25247744}a^{22}-\frac{7421}{16384}a^{20}-\frac{3083}{16384}a^{18}+\frac{4051}{16384}a^{16}+\frac{8101}{16384}a^{14}-\frac{457}{3155968}a^{12}-\frac{271}{788992}a^{10}-\frac{89}{197248}a^{8}+\frac{1}{49312}a^{6}+\frac{1}{536}a^{4}+\frac{17}{3082}a^{2}+\frac{14}{1541}$, $\frac{1}{50495488}a^{35}-\frac{1}{50495488}a^{33}-\frac{7}{50495488}a^{31}-\frac{17}{50495488}a^{29}-\frac{1}{2195456}a^{27}-\frac{1}{50495488}a^{25}+\frac{89}{50495488}a^{23}-\frac{7421}{32768}a^{21}-\frac{3083}{32768}a^{19}-\frac{12333}{32768}a^{17}+\frac{8101}{32768}a^{15}-\frac{457}{6311936}a^{13}-\frac{271}{1577984}a^{11}-\frac{89}{394496}a^{9}+\frac{1}{98624}a^{7}+\frac{1}{1072}a^{5}+\frac{17}{6164}a^{3}+\frac{7}{1541}a$, $\frac{1}{100990976}a^{36}-\frac{1}{100990976}a^{34}-\frac{7}{100990976}a^{32}-\frac{17}{100990976}a^{30}-\frac{1}{4390912}a^{28}-\frac{1}{100990976}a^{26}+\frac{89}{100990976}a^{24}+\frac{271}{100990976}a^{22}+\frac{29685}{65536}a^{20}-\frac{12333}{65536}a^{18}-\frac{24667}{65536}a^{16}-\frac{457}{12623872}a^{14}-\frac{271}{3155968}a^{12}-\frac{89}{788992}a^{10}+\frac{1}{197248}a^{8}+\frac{1}{2144}a^{6}+\frac{17}{12328}a^{4}+\frac{7}{3082}a^{2}+\frac{2}{1541}$, $\frac{1}{201981952}a^{37}-\frac{1}{201981952}a^{35}-\frac{7}{201981952}a^{33}-\frac{17}{201981952}a^{31}-\frac{1}{8781824}a^{29}-\frac{1}{201981952}a^{27}+\frac{89}{201981952}a^{25}+\frac{271}{201981952}a^{23}+\frac{29685}{131072}a^{21}-\frac{12333}{131072}a^{19}-\frac{24667}{131072}a^{17}-\frac{457}{25247744}a^{15}-\frac{271}{6311936}a^{13}-\frac{89}{1577984}a^{11}+\frac{1}{394496}a^{9}+\frac{1}{4288}a^{7}+\frac{17}{24656}a^{5}+\frac{7}{6164}a^{3}+\frac{1}{1541}a$, $\frac{1}{403963904}a^{38}-\frac{1}{403963904}a^{36}-\frac{7}{403963904}a^{34}-\frac{17}{403963904}a^{32}-\frac{1}{17563648}a^{30}-\frac{1}{403963904}a^{28}+\frac{89}{403963904}a^{26}+\frac{271}{403963904}a^{24}+\frac{457}{403963904}a^{22}+\frac{118739}{262144}a^{20}-\frac{24667}{262144}a^{18}-\frac{457}{50495488}a^{16}-\frac{271}{12623872}a^{14}-\frac{89}{3155968}a^{12}+\frac{1}{788992}a^{10}+\frac{1}{8576}a^{8}+\frac{17}{49312}a^{6}+\frac{7}{12328}a^{4}+\frac{1}{3082}a^{2}-\frac{2}{1541}$, $\frac{1}{807927808}a^{39}-\frac{1}{807927808}a^{37}-\frac{7}{807927808}a^{35}-\frac{17}{807927808}a^{33}-\frac{1}{35127296}a^{31}-\frac{1}{807927808}a^{29}+\frac{89}{807927808}a^{27}+\frac{271}{807927808}a^{25}+\frac{457}{807927808}a^{23}+\frac{118739}{524288}a^{21}+\frac{237477}{524288}a^{19}-\frac{457}{100990976}a^{17}-\frac{271}{25247744}a^{15}-\frac{89}{6311936}a^{13}+\frac{1}{1577984}a^{11}+\frac{1}{17152}a^{9}+\frac{17}{98624}a^{7}+\frac{7}{24656}a^{5}+\frac{1}{6164}a^{3}-\frac{1}{1541}a$ 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, 1/2*a^21 - 1/2*a^19 - 1/2*a^17 - 1/2*a^15 - 1/2*a^13 - 1/2*a^11 - 1/2*a^9 - 1/2*a^7 - 1/2*a^5 - 1/2*a^3 - 1/2*a, 1/6164*a^22 - 1/4*a^20 + 1/4*a^18 - 1/4*a^16 + 1/4*a^14 - 1/4*a^12 + 1/4*a^10 - 1/4*a^8 + 1/4*a^6 - 1/4*a^4 + 1/4*a^2 + 627/1541, 1/12328*a^23 - 1/8*a^21 + 1/8*a^19 - 1/8*a^17 + 1/8*a^15 - 1/8*a^13 + 1/8*a^11 - 1/8*a^9 + 1/8*a^7 - 1/8*a^5 + 1/8*a^3 + 627/3082*a, 1/24656*a^24 - 1/24656*a^22 + 5/16*a^20 + 3/16*a^18 + 5/16*a^16 + 3/16*a^14 + 5/16*a^12 + 3/16*a^10 + 5/16*a^8 + 3/16*a^6 + 5/16*a^4 - 457/3082*a^2 - 542/1541, 1/49312*a^25 - 1/49312*a^23 + 5/32*a^21 + 3/32*a^19 - 11/32*a^17 - 13/32*a^15 + 5/32*a^13 + 3/32*a^11 - 11/32*a^9 - 13/32*a^7 + 5/32*a^5 - 457/6164*a^3 - 271/1541*a, 1/98624*a^26 - 1/98624*a^24 - 7/98624*a^22 - 29/64*a^20 + 21/64*a^18 - 13/64*a^16 + 5/64*a^14 + 3/64*a^12 - 11/64*a^10 + 19/64*a^8 - 27/64*a^6 - 457/12328*a^4 - 271/3082*a^2 - 178/1541, 1/197248*a^27 - 1/197248*a^25 - 7/197248*a^23 - 29/128*a^21 + 21/128*a^19 + 51/128*a^17 - 59/128*a^15 + 3/128*a^13 - 11/128*a^11 - 45/128*a^9 + 37/128*a^7 - 457/24656*a^5 - 271/6164*a^3 - 89/1541*a, 1/394496*a^28 - 1/394496*a^26 - 7/394496*a^24 - 17/394496*a^22 - 107/256*a^20 + 51/256*a^18 + 69/256*a^16 + 3/256*a^14 - 11/256*a^12 - 45/256*a^10 - 91/256*a^8 - 457/49312*a^6 - 271/12328*a^4 - 89/3082*a^2 + 2/1541, 1/788992*a^29 - 1/788992*a^27 - 7/788992*a^25 - 17/788992*a^23 - 107/512*a^21 - 205/512*a^19 - 187/512*a^17 - 253/512*a^15 - 11/512*a^13 - 45/512*a^11 - 91/512*a^9 - 457/98624*a^7 - 271/24656*a^5 - 89/6164*a^3 + 1/1541*a, 1/1577984*a^30 - 1/1577984*a^28 - 7/1577984*a^26 - 17/1577984*a^24 - 1/68608*a^22 + 307/1024*a^20 + 325/1024*a^18 - 253/1024*a^16 - 11/1024*a^14 - 45/1024*a^12 - 91/1024*a^10 - 457/197248*a^8 - 271/49312*a^6 - 89/12328*a^4 + 1/3082*a^2 + 2/67, 1/3155968*a^31 - 1/3155968*a^29 - 7/3155968*a^27 - 17/3155968*a^25 - 1/137216*a^23 + 307/2048*a^21 - 699/2048*a^19 + 771/2048*a^17 + 1013/2048*a^15 - 45/2048*a^13 - 91/2048*a^11 - 457/394496*a^9 - 271/98624*a^7 - 89/24656*a^5 + 1/6164*a^3 + 1/67*a, 1/6311936*a^32 - 1/6311936*a^30 - 7/6311936*a^28 - 17/6311936*a^26 - 1/274432*a^24 - 1/6311936*a^22 - 699/4096*a^20 + 771/4096*a^18 + 1013/4096*a^16 - 45/4096*a^14 - 91/4096*a^12 - 457/788992*a^10 - 271/197248*a^8 - 89/49312*a^6 + 1/12328*a^4 + 1/134*a^2 + 34/1541, 1/12623872*a^33 - 1/12623872*a^31 - 7/12623872*a^29 - 17/12623872*a^27 - 1/548864*a^25 - 1/12623872*a^23 - 699/8192*a^21 - 3325/8192*a^19 + 1013/8192*a^17 - 45/8192*a^15 + 4005/8192*a^13 + 788535/1577984*a^11 + 196977/394496*a^9 + 49223/98624*a^7 - 12327/24656*a^5 - 133/268*a^3 - 1507/3082*a, 1/25247744*a^34 - 1/25247744*a^32 - 7/25247744*a^30 - 17/25247744*a^28 - 1/1097728*a^26 - 1/25247744*a^24 + 89/25247744*a^22 - 7421/16384*a^20 - 3083/16384*a^18 + 4051/16384*a^16 + 8101/16384*a^14 - 457/3155968*a^12 - 271/788992*a^10 - 89/197248*a^8 + 1/49312*a^6 + 1/536*a^4 + 17/3082*a^2 + 14/1541, 1/50495488*a^35 - 1/50495488*a^33 - 7/50495488*a^31 - 17/50495488*a^29 - 1/2195456*a^27 - 1/50495488*a^25 + 89/50495488*a^23 - 7421/32768*a^21 - 3083/32768*a^19 - 12333/32768*a^17 + 8101/32768*a^15 - 457/6311936*a^13 - 271/1577984*a^11 - 89/394496*a^9 + 1/98624*a^7 + 1/1072*a^5 + 17/6164*a^3 + 7/1541*a, 1/100990976*a^36 - 1/100990976*a^34 - 7/100990976*a^32 - 17/100990976*a^30 - 1/4390912*a^28 - 1/100990976*a^26 + 89/100990976*a^24 + 271/100990976*a^22 + 29685/65536*a^20 - 12333/65536*a^18 - 24667/65536*a^16 - 457/12623872*a^14 - 271/3155968*a^12 - 89/788992*a^10 + 1/197248*a^8 + 1/2144*a^6 + 17/12328*a^4 + 7/3082*a^2 + 2/1541, 1/201981952*a^37 - 1/201981952*a^35 - 7/201981952*a^33 - 17/201981952*a^31 - 1/8781824*a^29 - 1/201981952*a^27 + 89/201981952*a^25 + 271/201981952*a^23 + 29685/131072*a^21 - 12333/131072*a^19 - 24667/131072*a^17 - 457/25247744*a^15 - 271/6311936*a^13 - 89/1577984*a^11 + 1/394496*a^9 + 1/4288*a^7 + 17/24656*a^5 + 7/6164*a^3 + 1/1541*a, 1/403963904*a^38 - 1/403963904*a^36 - 7/403963904*a^34 - 17/403963904*a^32 - 1/17563648*a^30 - 1/403963904*a^28 + 89/403963904*a^26 + 271/403963904*a^24 + 457/403963904*a^22 + 118739/262144*a^20 - 24667/262144*a^18 - 457/50495488*a^16 - 271/12623872*a^14 - 89/3155968*a^12 + 1/788992*a^10 + 1/8576*a^8 + 17/49312*a^6 + 7/12328*a^4 + 1/3082*a^2 - 2/1541, 1/807927808*a^39 - 1/807927808*a^37 - 7/807927808*a^35 - 17/807927808*a^33 - 1/35127296*a^31 - 1/807927808*a^29 + 89/807927808*a^27 + 271/807927808*a^25 + 457/807927808*a^23 + 118739/524288*a^21 + 237477/524288*a^19 - 457/100990976*a^17 - 271/25247744*a^15 - 89/6311936*a^13 + 1/1577984*a^11 + 1/17152*a^9 + 17/98624*a^7 + 7/24656*a^5 + 1/6164*a^3 - 1/1541*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{1}{49312} a^{27} + \frac{8279}{49312} a^{5} $ -(1)/(49312)a^(27) + (8279)/(49312)a^(5) (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 + 3*x^38 + 5*x^36 + 3*x^34 - 11*x^32 - 45*x^30 - 91*x^28 - 93*x^26 + 85*x^24 + 627*x^22 + 1541*x^20 + 2508*x^18 + 1360*x^16 - 5952*x^14 - 23296*x^12 - 46080*x^10 - 45056*x^8 + 49152*x^6 + 327680*x^4 + 786432*x^2 + 1048576)

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 + 3*x^38 + 5*x^36 + 3*x^34 - 11*x^32 - 45*x^30 - 91*x^28 - 93*x^26 + 85*x^24 + 627*x^22 + 1541*x^20 + 2508*x^18 + 1360*x^16 - 5952*x^14 - 23296*x^12 - 46080*x^10 - 45056*x^8 + 49152*x^6 + 327680*x^4 + 786432*x^2 + 1048576, 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 + 3*x^38 + 5*x^36 + 3*x^34 - 11*x^32 - 45*x^30 - 91*x^28 - 93*x^26 + 85*x^24 + 627*x^22 + 1541*x^20 + 2508*x^18 + 1360*x^16 - 5952*x^14 - 23296*x^12 - 46080*x^10 - 45056*x^8 + 49152*x^6 + 327680*x^4 + 786432*x^2 + 1048576);

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 + 3*x^38 + 5*x^36 + 3*x^34 - 11*x^32 - 45*x^30 - 91*x^28 - 93*x^26 + 85*x^24 + 627*x^22 + 1541*x^20 + 2508*x^18 + 1360*x^16 - 5952*x^14 - 23296*x^12 - 46080*x^10 - 45056*x^8 + 49152*x^6 + 327680*x^4 + 786432*x^2 + 1048576);

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 $C_2^2\times C_{10}$ is not computed

Intermediate fields

$\Q(\sqrt{-1}) $, $\Q(\sqrt{77}) $, $\Q(\sqrt{-77}) $, $\Q(\sqrt{-7}) $, $\Q(\sqrt{7}) $, $\Q(\sqrt{-11}) $, $\Q(\sqrt{11}) $, $\Q(i, \sqrt{77})$, $\Q(i, \sqrt{7})$, $\Q(i, \sqrt{11})$, $\Q(\sqrt{-7}, \sqrt{-11})$, $\Q(\sqrt{7}, \sqrt{11})$, $\Q(\sqrt{-7}, \sqrt{11})$, $\Q(\sqrt{7}, \sqrt{-11})$, $\Q(\zeta_{11})^+$, 8.0.8999178496.1, 10.0.219503494144.1, 10.10.39630026842637.1, 10.0.40581147486860288.1, 10.0.3602729712967.1, 10.10.3689195226078208.1, $\Q(\zeta_{11})$, $\Q(\zeta_{44})^+$, 20.0.1646829531350307068613612031442944.1, 20.0.13610161416118240236476132491264.1, $\Q(\zeta_{44})$, 20.0.1570539027548129147161113769.2, 20.20.1646829531350307068613612031442944.1, 20.0.1646829531350307068613612031442944.3, 20.0.1646829531350307068613612031442944.4

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	${\href{/padicField/3.10.0.1}{10} }^{4}$	${\href{/padicField/5.10.0.1}{10} }^{4}$	R	R	${\href{/padicField/13.10.0.1}{10} }^{4}$	${\href{/padicField/17.10.0.1}{10} }^{4}$	${\href{/padicField/19.10.0.1}{10} }^{4}$	${\href{/padicField/23.2.0.1}{2} }^{20}$	${\href{/padicField/29.10.0.1}{10} }^{4}$	${\href{/padicField/31.10.0.1}{10} }^{4}$	${\href{/padicField/37.5.0.1}{5} }^{8}$	${\href{/padicField/41.10.0.1}{10} }^{4}$	${\href{/padicField/43.2.0.1}{2} }^{20}$	${\href{/padicField/47.10.0.1}{10} }^{4}$	${\href{/padicField/53.5.0.1}{5} }^{8}$	${\href{/padicField/59.10.0.1}{10} }^{4}$

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 $20$	$2$	$10$	$20$
$7$ 7	7.20.10.1	$x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$	$2$	$10$	$10$	20T3	$[\ ]_{2}^{10}$
$7$ 7	7.20.10.1	$x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$	$2$	$10$	$10$	20T3	$[\ ]_{2}^{10}$
$11$ 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	$[\ ]_{10}^{2}$

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