Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^42 - 84*x^40 + 3276*x^38 - 78736*x^36 + 1305360*x^34 - 15833664*x^32 + 145435136*x^30 - 1032886144*x^28 + 5741625344*x^26 - 25131545600*x^24 + 86701812736*x^22 - 234915702784*x^20 + 495818960896*x^18 - 804378279936*x^16 + 983365402624*x^14 - 880319397888*x^12 + 553414557696*x^10 - 229269569536*x^8 + 56479711232*x^6 - 6910640128*x^4 + 308281344*x^2 - 2097152)
gp: K = bnfinit(y^42 - 84*y^40 + 3276*y^38 - 78736*y^36 + 1305360*y^34 - 15833664*y^32 + 145435136*y^30 - 1032886144*y^28 + 5741625344*y^26 - 25131545600*y^24 + 86701812736*y^22 - 234915702784*y^20 + 495818960896*y^18 - 804378279936*y^16 + 983365402624*y^14 - 880319397888*y^12 + 553414557696*y^10 - 229269569536*y^8 + 56479711232*y^6 - 6910640128*y^4 + 308281344*y^2 - 2097152, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^42 - 84*x^40 + 3276*x^38 - 78736*x^36 + 1305360*x^34 - 15833664*x^32 + 145435136*x^30 - 1032886144*x^28 + 5741625344*x^26 - 25131545600*x^24 + 86701812736*x^22 - 234915702784*x^20 + 495818960896*x^18 - 804378279936*x^16 + 983365402624*x^14 - 880319397888*x^12 + 553414557696*x^10 - 229269569536*x^8 + 56479711232*x^6 - 6910640128*x^4 + 308281344*x^2 - 2097152);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^42 - 84*x^40 + 3276*x^38 - 78736*x^36 + 1305360*x^34 - 15833664*x^32 + 145435136*x^30 - 1032886144*x^28 + 5741625344*x^26 - 25131545600*x^24 + 86701812736*x^22 - 234915702784*x^20 + 495818960896*x^18 - 804378279936*x^16 + 983365402624*x^14 - 880319397888*x^12 + 553414557696*x^10 - 229269569536*x^8 + 56479711232*x^6 - 6910640128*x^4 + 308281344*x^2 - 2097152)
\( x^{42} - 84 x^{40} + 3276 x^{38} - 78736 x^{36} + 1305360 x^{34} - 15833664 x^{32} + 145435136 x^{30} + \cdots - 2097152 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $42$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[42, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(155\!\cdots\!608\)
\(\medspace = 2^{63}\cdot 7^{76}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(95.67\) | 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: | $2^{3/2}7^{38/21}\approx 95.66873209957993$ | ||
| Ramified primes: |
\(2\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $42$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(392=2^{3}\cdot 7^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{392}(1,·)$, $\chi_{392}(261,·)$, $\chi_{392}(65,·)$, $\chi_{392}(9,·)$, $\chi_{392}(141,·)$, $\chi_{392}(365,·)$, $\chi_{392}(277,·)$, $\chi_{392}(25,·)$, $\chi_{392}(281,·)$, $\chi_{392}(29,·)$, $\chi_{392}(389,·)$, $\chi_{392}(289,·)$, $\chi_{392}(37,·)$, $\chi_{392}(305,·)$, $\chi_{392}(169,·)$, $\chi_{392}(93,·)$, $\chi_{392}(177,·)$, $\chi_{392}(53,·)$, $\chi_{392}(137,·)$, $\chi_{392}(57,·)$, $\chi_{392}(317,·)$, $\chi_{392}(309,·)$, $\chi_{392}(193,·)$, $\chi_{392}(197,·)$, $\chi_{392}(333,·)$, $\chi_{392}(205,·)$, $\chi_{392}(337,·)$, $\chi_{392}(85,·)$, $\chi_{392}(121,·)$, $\chi_{392}(345,·)$, $\chi_{392}(221,·)$, $\chi_{392}(165,·)$, $\chi_{392}(225,·)$, $\chi_{392}(81,·)$, $\chi_{392}(361,·)$, $\chi_{392}(109,·)$, $\chi_{392}(113,·)$, $\chi_{392}(373,·)$, $\chi_{392}(233,·)$, $\chi_{392}(249,·)$, $\chi_{392}(253,·)$, $\chi_{392}(149,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$, $\frac{1}{16384}a^{28}$, $\frac{1}{16384}a^{29}$, $\frac{1}{32768}a^{30}$, $\frac{1}{32768}a^{31}$, $\frac{1}{65536}a^{32}$, $\frac{1}{65536}a^{33}$, $\frac{1}{131072}a^{34}$, $\frac{1}{131072}a^{35}$, $\frac{1}{262144}a^{36}$, $\frac{1}{262144}a^{37}$, $\frac{1}{524288}a^{38}$, $\frac{1}{524288}a^{39}$, $\frac{1}{1048576}a^{40}$, $\frac{1}{1048576}a^{41}$
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
Trivial group, which has order $1$ (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: | $41$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{1}{16384}a^{28}-\frac{7}{2048}a^{26}+\frac{175}{2048}a^{24}-\frac{161}{128}a^{22}+\frac{12397}{1024}a^{20}-\frac{10241}{128}a^{18}+\frac{47481}{128}a^{16}-\frac{155039}{128}a^{14}+\frac{22043}{8}a^{12}-\frac{136059}{32}a^{10}+\frac{33929}{8}a^{8}-\frac{10045}{4}a^{6}+\frac{2989}{4}a^{4}-\frac{147}{2}a^{2}+1$, $\frac{1}{128}a^{14}-\frac{7}{32}a^{12}+\frac{77}{32}a^{10}-\frac{105}{8}a^{8}+\frac{147}{4}a^{6}-49a^{4}+\frac{49}{2}a^{2}-2$, $\frac{1}{131072}a^{34}-\frac{17}{32768}a^{32}+\frac{527}{32768}a^{30}-\frac{2465}{8192}a^{28}+\frac{31059}{8192}a^{26}-\frac{69615}{2048}a^{24}+\frac{228735}{1024}a^{22}-\frac{279565}{256}a^{20}+\frac{2042975}{512}a^{18}-\frac{1389223}{128}a^{16}+\frac{1389223}{64}a^{14}-\frac{499681}{16}a^{12}+\frac{499681}{16}a^{10}-\frac{82365}{4}a^{8}+\frac{16473}{2}a^{6}-1734a^{4}+\frac{289}{2}a^{2}-2$, $\frac{1}{32768}a^{30}-\frac{15}{8192}a^{28}+\frac{405}{8192}a^{26}-\frac{1625}{2048}a^{24}+\frac{8625}{1024}a^{22}-\frac{15939}{256}a^{20}+\frac{168245}{512}a^{18}-\frac{159885}{128}a^{16}+\frac{218025}{64}a^{14}-\frac{104975}{16}a^{12}+\frac{138567}{16}a^{10}-\frac{29835}{4}a^{8}+\frac{7735}{2}a^{6}-1050a^{4}+\frac{225}{2}a^{2}-2$, $\frac{1}{65536}a^{32}-\frac{1}{1024}a^{30}+\frac{29}{1024}a^{28}-\frac{63}{128}a^{26}+\frac{2925}{512}a^{24}-\frac{1495}{32}a^{22}+\frac{8855}{32}a^{20}-\frac{4807}{4}a^{18}+\frac{245157}{64}a^{16}-\frac{17765}{2}a^{14}+\frac{29393}{2}a^{12}-16796a^{10}+12597a^{8}-5712a^{6}+1360a^{4}-128a^{2}+2$, $\frac{1}{524288}a^{38}-\frac{19}{131072}a^{36}+\frac{665}{131072}a^{34}-\frac{3553}{32768}a^{32}+\frac{6479}{4096}a^{30}-\frac{17081}{1024}a^{28}+\frac{1076103}{8192}a^{26}-\frac{1611675}{2048}a^{24}+\frac{7413705}{2048}a^{22}-\frac{6561555}{512}a^{20}+\frac{17809935}{512}a^{18}-\frac{9174815}{128}a^{16}+\frac{3528775}{32}a^{14}-\frac{988057}{8}a^{12}+\frac{1552661}{16}a^{10}-\frac{202521}{4}a^{8}+\frac{128877}{8}a^{6}-\frac{5415}{2}a^{4}+\frac{361}{2}a^{2}-2$, $\frac{1}{2048}a^{22}-\frac{11}{512}a^{20}+\frac{209}{512}a^{18}-\frac{561}{128}a^{16}+\frac{935}{32}a^{14}-\frac{1001}{8}a^{12}+\frac{11011}{32}a^{10}-\frac{4719}{8}a^{8}+\frac{4719}{8}a^{6}-\frac{605}{2}a^{4}+\frac{121}{2}a^{2}-2$, $\frac{1}{1048576}a^{40}-\frac{5}{65536}a^{38}+\frac{185}{65536}a^{36}-\frac{525}{8192}a^{34}+\frac{32725}{32768}a^{32}-\frac{371007}{32768}a^{30}+\frac{791105}{8192}a^{28}-\frac{2588917}{4096}a^{26}+\frac{13144599}{4096}a^{24}-\frac{25995451}{2048}a^{22}+\frac{19997131}{512}a^{20}-\frac{23771755}{256}a^{18}+\frac{43114687}{256}a^{16}-\frac{14614789}{64}a^{14}+\frac{14369253}{64}a^{12}-\frac{611255}{4}a^{10}+\frac{1069979}{16}a^{8}-\frac{132391}{8}a^{6}+\frac{7175}{4}a^{4}-\frac{105}{2}a^{2}$, $\frac{1}{32}a^{10}-\frac{5}{8}a^{8}+\frac{35}{8}a^{6}-\frac{25}{2}a^{4}+\frac{25}{2}a^{2}-2$, $\frac{1}{1024}a^{20}-\frac{5}{128}a^{18}+\frac{85}{128}a^{16}-\frac{25}{4}a^{14}+\frac{2275}{64}a^{12}-\frac{1001}{8}a^{10}+\frac{2145}{8}a^{8}-330a^{6}+\frac{825}{4}a^{4}-50a^{2}+2$, $\frac{1}{1048576}a^{40}-\frac{5}{65536}a^{38}+\frac{185}{65536}a^{36}-\frac{525}{8192}a^{34}+\frac{32725}{32768}a^{32}-\frac{5797}{512}a^{30}+\frac{49445}{512}a^{28}-\frac{40455}{64}a^{26}+\frac{13147875}{4096}a^{24}-\frac{3251625}{256}a^{22}+\frac{10015005}{256}a^{20}-\frac{2982525}{32}a^{18}+\frac{21729825}{128}a^{16}-\frac{928625}{4}a^{14}+\frac{928625}{4}a^{12}-163438a^{10}+\frac{1225785}{16}a^{8}-21945a^{6}+3325a^{4}-200a^{2}+2$, $\frac{1}{512}a^{18}-\frac{9}{128}a^{16}+\frac{135}{128}a^{14}-\frac{273}{32}a^{12}+\frac{1287}{32}a^{10}-\frac{891}{8}a^{8}+\frac{693}{4}a^{6}-135a^{4}+\frac{81}{2}a^{2}-2$, $\frac{1}{262144}a^{36}-\frac{9}{32768}a^{34}+\frac{297}{32768}a^{32}-\frac{93}{512}a^{30}+\frac{40455}{16384}a^{28}-\frac{49329}{2048}a^{26}+\frac{356265}{2048}a^{24}-\frac{121095}{128}a^{22}+\frac{3996135}{1024}a^{20}-\frac{1562275}{128}a^{18}+\frac{3677355}{128}a^{16}-\frac{200583}{4}a^{14}+\frac{2028117}{32}a^{12}-\frac{223839}{4}a^{10}+\frac{130815}{4}a^{8}-11628a^{6}+\frac{8721}{4}a^{4}-162a^{2}+2$, $\frac{1}{8192}a^{26}-\frac{13}{2048}a^{24}+\frac{299}{2048}a^{22}-\frac{1001}{512}a^{20}+\frac{8645}{512}a^{18}-\frac{12597}{128}a^{16}+\frac{12597}{32}a^{14}-\frac{8619}{8}a^{12}+\frac{31603}{16}a^{10}-\frac{9295}{4}a^{8}+\frac{13013}{8}a^{6}-\frac{1183}{2}a^{4}+\frac{169}{2}a^{2}-2$, $\frac{1}{524288}a^{38}-\frac{19}{131072}a^{36}+\frac{665}{131072}a^{34}-\frac{7105}{65536}a^{32}+\frac{6475}{4096}a^{30}-\frac{4263}{256}a^{28}+\frac{1072071}{8192}a^{26}-\frac{3199949}{4096}a^{24}+\frac{7318001}{2048}a^{22}-\frac{6419749}{512}a^{20}+\frac{537285}{16}a^{18}-\frac{8681603}{128}a^{16}+\frac{12963587}{128}a^{14}-\frac{3469837}{32}a^{12}+\frac{1270841}{16}a^{10}-\frac{591019}{16}a^{8}+\frac{38297}{4}a^{6}-\frac{4263}{4}a^{4}+\frac{63}{2}a^{2}$, $\frac{1}{4}a^{4}-2a^{2}+2$, $\frac{1}{8}a^{6}-\frac{3}{2}a^{4}+\frac{9}{2}a^{2}-2$, $\frac{1}{16}a^{8}-a^{6}+5a^{4}-8a^{2}+2$, $\frac{1}{256}a^{16}-\frac{1}{8}a^{14}+\frac{13}{8}a^{12}-11a^{10}+\frac{165}{4}a^{8}-84a^{6}+84a^{4}-32a^{2}+2$, $\frac{1}{64}a^{12}-\frac{3}{8}a^{10}+\frac{27}{8}a^{8}-14a^{6}+\frac{105}{4}a^{4}-18a^{2}+2$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}+\frac{35}{2048}a^{31}-\frac{5425}{16384}a^{29}-\frac{1}{16384}a^{28}+\frac{35525}{8192}a^{27}+\frac{7}{2048}a^{26}-\frac{166257}{4096}a^{25}-\frac{175}{2048}a^{24}+\frac{143325}{512}a^{23}+\frac{161}{128}a^{22}-\frac{1480049}{1024}a^{21}-\frac{12397}{1024}a^{20}+\frac{1438927}{256}a^{19}+\frac{10241}{128}a^{18}-\frac{262871}{16}a^{17}-\frac{47481}{128}a^{16}+\frac{285957}{8}a^{15}+\frac{4845}{4}a^{14}-\frac{1818635}{32}a^{13}-\frac{88179}{32}a^{12}+\frac{2051777}{32}a^{11}+\frac{17017}{4}a^{10}-\frac{784343}{16}a^{9}-\frac{17017}{4}a^{8}+\frac{188653}{8}a^{7}+2548a^{6}-6265a^{5}-\frac{3185}{4}a^{4}+707a^{3}+98a^{2}-21a-2$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}+\frac{35}{2048}a^{31}-\frac{5425}{16384}a^{29}+\frac{35525}{8192}a^{27}-\frac{166257}{4096}a^{25}+\frac{143325}{512}a^{23}-\frac{1480049}{1024}a^{21}+\frac{1438927}{256}a^{19}-\frac{262871}{16}a^{17}+\frac{285957}{8}a^{15}+\frac{1}{128}a^{14}-\frac{1818635}{32}a^{13}-\frac{7}{32}a^{12}+\frac{2051777}{32}a^{11}+\frac{77}{32}a^{10}-\frac{784343}{16}a^{9}-\frac{105}{8}a^{8}+\frac{188653}{8}a^{7}+\frac{147}{4}a^{6}-6265a^{5}-49a^{4}+707a^{3}+\frac{49}{2}a^{2}-21a-2$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}-\frac{1}{65536}a^{32}+\frac{35}{2048}a^{31}+\frac{1}{1024}a^{30}-\frac{5425}{16384}a^{29}-\frac{29}{1024}a^{28}+\frac{35525}{8192}a^{27}+\frac{63}{128}a^{26}-\frac{166257}{4096}a^{25}-\frac{2925}{512}a^{24}+\frac{143325}{512}a^{23}+\frac{1495}{32}a^{22}-\frac{1480049}{1024}a^{21}-\frac{8855}{32}a^{20}+\frac{1438927}{256}a^{19}+\frac{4807}{4}a^{18}-\frac{262871}{16}a^{17}-\frac{245157}{64}a^{16}+\frac{285957}{8}a^{15}+\frac{17765}{2}a^{14}-\frac{1818635}{32}a^{13}-\frac{29393}{2}a^{12}+\frac{2051777}{32}a^{11}+16796a^{10}-\frac{784343}{16}a^{9}-12597a^{8}+\frac{188653}{8}a^{7}+5712a^{6}-6265a^{5}-1360a^{4}+707a^{3}+128a^{2}-21a-2$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}+\frac{35}{2048}a^{31}-\frac{5425}{16384}a^{29}+\frac{35525}{8192}a^{27}-\frac{166257}{4096}a^{25}+\frac{143325}{512}a^{23}-\frac{1}{2048}a^{22}-\frac{1480049}{1024}a^{21}+\frac{11}{512}a^{20}+\frac{1438927}{256}a^{19}-\frac{209}{512}a^{18}-\frac{262871}{16}a^{17}+\frac{561}{128}a^{16}+\frac{285957}{8}a^{15}-\frac{935}{32}a^{14}-\frac{1818635}{32}a^{13}+\frac{1001}{8}a^{12}+\frac{2051777}{32}a^{11}-\frac{11011}{32}a^{10}-\frac{784343}{16}a^{9}+\frac{4719}{8}a^{8}+\frac{188653}{8}a^{7}-\frac{4719}{8}a^{6}-6265a^{5}+\frac{605}{2}a^{4}+707a^{3}-\frac{121}{2}a^{2}-21a+2$, $\frac{1}{262144}a^{36}+\frac{1}{131072}a^{35}-\frac{9}{32768}a^{34}-\frac{35}{65536}a^{33}+\frac{297}{32768}a^{32}+\frac{35}{2048}a^{31}-\frac{93}{512}a^{30}-\frac{5425}{16384}a^{29}+\frac{40455}{16384}a^{28}+\frac{35525}{8192}a^{27}-\frac{49329}{2048}a^{26}-\frac{166257}{4096}a^{25}+\frac{356265}{2048}a^{24}+\frac{143325}{512}a^{23}-\frac{121095}{128}a^{22}-\frac{1480049}{1024}a^{21}+\frac{3996135}{1024}a^{20}+\frac{1438927}{256}a^{19}-\frac{1562275}{128}a^{18}-\frac{262871}{16}a^{17}+\frac{3677355}{128}a^{16}+\frac{285957}{8}a^{15}-\frac{200583}{4}a^{14}-\frac{1818635}{32}a^{13}+\frac{2028117}{32}a^{12}+\frac{2051777}{32}a^{11}-\frac{223839}{4}a^{10}-\frac{784343}{16}a^{9}+\frac{130815}{4}a^{8}+\frac{188653}{8}a^{7}-11628a^{6}-6265a^{5}+\frac{8721}{4}a^{4}+707a^{3}-162a^{2}-21a+2$, $\frac{1}{262144}a^{36}+\frac{1}{131072}a^{35}-\frac{35}{131072}a^{34}-\frac{35}{65536}a^{33}+\frac{35}{4096}a^{32}+\frac{35}{2048}a^{31}-\frac{5425}{32768}a^{30}-\frac{5425}{16384}a^{29}+\frac{35525}{16384}a^{28}+\frac{35525}{8192}a^{27}-\frac{166257}{8192}a^{26}-\frac{166257}{4096}a^{25}+\frac{143325}{1024}a^{24}+\frac{143325}{512}a^{23}-\frac{1480049}{2048}a^{22}-\frac{1480049}{1024}a^{21}+\frac{1438927}{512}a^{20}+\frac{1438927}{256}a^{19}-\frac{262871}{32}a^{18}-\frac{262871}{16}a^{17}+\frac{285957}{16}a^{16}+\frac{285957}{8}a^{15}-\frac{1818635}{64}a^{14}-\frac{1818635}{32}a^{13}+\frac{2051777}{64}a^{12}+\frac{2051777}{32}a^{11}-\frac{784343}{32}a^{10}-\frac{784343}{16}a^{9}+\frac{188653}{16}a^{8}+\frac{188653}{8}a^{7}-\frac{6265}{2}a^{6}-6265a^{5}+\frac{707}{2}a^{4}+707a^{3}-\frac{21}{2}a^{2}-21a$, $\frac{1}{131072}a^{35}-\frac{1}{131072}a^{34}-\frac{35}{65536}a^{33}+\frac{17}{32768}a^{32}+\frac{35}{2048}a^{31}-\frac{527}{32768}a^{30}-\frac{5425}{16384}a^{29}+\frac{2465}{8192}a^{28}+\frac{35525}{8192}a^{27}-\frac{31059}{8192}a^{26}-\frac{166257}{4096}a^{25}+\frac{69615}{2048}a^{24}+\frac{143325}{512}a^{23}-\frac{228735}{1024}a^{22}-\frac{1480049}{1024}a^{21}+\frac{279565}{256}a^{20}+\frac{1438927}{256}a^{19}-\frac{2042975}{512}a^{18}-\frac{262871}{16}a^{17}+\frac{1389223}{128}a^{16}+\frac{285957}{8}a^{15}-\frac{1389223}{64}a^{14}-\frac{1818635}{32}a^{13}+\frac{499681}{16}a^{12}+\frac{2051777}{32}a^{11}-\frac{499681}{16}a^{10}-\frac{784343}{16}a^{9}+\frac{82365}{4}a^{8}+\frac{188653}{8}a^{7}-\frac{16473}{2}a^{6}-6265a^{5}+1734a^{4}+707a^{3}-\frac{289}{2}a^{2}-21a+2$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}+\frac{35}{2048}a^{31}-\frac{5425}{16384}a^{29}+\frac{35525}{8192}a^{27}-\frac{166257}{4096}a^{25}+\frac{143325}{512}a^{23}-\frac{1480049}{1024}a^{21}+\frac{1438927}{256}a^{19}-\frac{262871}{16}a^{17}+\frac{285957}{8}a^{15}-\frac{1818635}{32}a^{13}+\frac{2051777}{32}a^{11}-\frac{784343}{16}a^{9}-\frac{1}{16}a^{8}+\frac{188653}{8}a^{7}+a^{6}-6265a^{5}-5a^{4}+707a^{3}+8a^{2}-21a-2$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}+\frac{35}{2048}a^{31}-\frac{5425}{16384}a^{29}+\frac{35525}{8192}a^{27}-\frac{166257}{4096}a^{25}-\frac{1}{4096}a^{24}+\frac{143325}{512}a^{23}+\frac{3}{256}a^{22}-\frac{1480049}{1024}a^{21}-\frac{63}{256}a^{20}+\frac{1438927}{256}a^{19}+\frac{95}{32}a^{18}-\frac{262871}{16}a^{17}-\frac{2907}{128}a^{16}+\frac{285957}{8}a^{15}+\frac{459}{4}a^{14}-\frac{1818635}{32}a^{13}-\frac{1547}{4}a^{12}+\frac{2051777}{32}a^{11}+858a^{10}-\frac{784343}{16}a^{9}-\frac{19305}{16}a^{8}+\frac{188653}{8}a^{7}+1001a^{6}-6265a^{5}-429a^{4}+707a^{3}+72a^{2}-21a-2$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}+\frac{35}{2048}a^{31}-\frac{5425}{16384}a^{29}+\frac{35525}{8192}a^{27}-\frac{166257}{4096}a^{25}+\frac{143325}{512}a^{23}-\frac{1480049}{1024}a^{21}+\frac{1438927}{256}a^{19}-\frac{262871}{16}a^{17}+\frac{285957}{8}a^{15}-\frac{1818635}{32}a^{13}+\frac{2051777}{32}a^{11}+\frac{1}{32}a^{10}-\frac{784343}{16}a^{9}-\frac{5}{8}a^{8}+\frac{188653}{8}a^{7}+\frac{35}{8}a^{6}-6265a^{5}-\frac{25}{2}a^{4}+707a^{3}+\frac{25}{2}a^{2}-21a-2$, $\frac{1}{524288}a^{38}-\frac{19}{131072}a^{36}+\frac{1}{131072}a^{35}+\frac{665}{131072}a^{34}-\frac{35}{65536}a^{33}-\frac{7105}{65536}a^{32}+\frac{35}{2048}a^{31}+\frac{6475}{4096}a^{30}-\frac{5425}{16384}a^{29}-\frac{4263}{256}a^{28}+\frac{35525}{8192}a^{27}+\frac{1072071}{8192}a^{26}-\frac{166257}{4096}a^{25}-\frac{3199949}{4096}a^{24}+\frac{143325}{512}a^{23}+\frac{7318001}{2048}a^{22}-\frac{1480049}{1024}a^{21}-\frac{6419749}{512}a^{20}+\frac{1438927}{256}a^{19}+\frac{537285}{16}a^{18}-\frac{262871}{16}a^{17}-\frac{8681603}{128}a^{16}+\frac{285957}{8}a^{15}+\frac{12963587}{128}a^{14}-\frac{1818635}{32}a^{13}-\frac{3469837}{32}a^{12}+\frac{2051777}{32}a^{11}+\frac{1270841}{16}a^{10}-\frac{784343}{16}a^{9}-\frac{591019}{16}a^{8}+\frac{188653}{8}a^{7}+\frac{38297}{4}a^{6}-6265a^{5}-\frac{4263}{4}a^{4}+707a^{3}+\frac{63}{2}a^{2}-21a$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}+\frac{35}{2048}a^{31}-\frac{5425}{16384}a^{29}+\frac{35525}{8192}a^{27}-\frac{166257}{4096}a^{25}+\frac{143325}{512}a^{23}-\frac{1480049}{1024}a^{21}+\frac{1438927}{256}a^{19}-\frac{262871}{16}a^{17}+\frac{285957}{8}a^{15}-\frac{1818635}{32}a^{13}+\frac{2051777}{32}a^{11}-\frac{784343}{16}a^{9}+\frac{188653}{8}a^{7}-6265a^{5}+\frac{1}{4}a^{4}+707a^{3}-2a^{2}-21a+2$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}+\frac{35}{2048}a^{31}-\frac{5425}{16384}a^{29}+\frac{35525}{8192}a^{27}-\frac{166257}{4096}a^{25}+\frac{143325}{512}a^{23}-\frac{1480049}{1024}a^{21}+\frac{1438927}{256}a^{19}-\frac{1}{512}a^{18}-\frac{262871}{16}a^{17}+\frac{9}{128}a^{16}+\frac{285957}{8}a^{15}-\frac{135}{128}a^{14}-\frac{1818635}{32}a^{13}+\frac{273}{32}a^{12}+\frac{2051777}{32}a^{11}-\frac{1287}{32}a^{10}-\frac{784343}{16}a^{9}+\frac{891}{8}a^{8}+\frac{188653}{8}a^{7}-\frac{693}{4}a^{6}-6265a^{5}+135a^{4}+707a^{3}-\frac{81}{2}a^{2}-21a+2$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}+\frac{35}{2048}a^{31}-\frac{5425}{16384}a^{29}+\frac{35525}{8192}a^{27}-\frac{166257}{4096}a^{25}+\frac{143325}{512}a^{23}-\frac{1480049}{1024}a^{21}+\frac{1438927}{256}a^{19}-\frac{262871}{16}a^{17}+\frac{285957}{8}a^{15}-\frac{1818635}{32}a^{13}+\frac{2051777}{32}a^{11}-\frac{784343}{16}a^{9}+\frac{188653}{8}a^{7}+\frac{1}{8}a^{6}-6265a^{5}-\frac{3}{2}a^{4}+707a^{3}+\frac{9}{2}a^{2}-21a-2$, $\frac{1}{1048576}a^{40}-\frac{5}{65536}a^{38}+\frac{185}{65536}a^{36}+\frac{1}{131072}a^{35}-\frac{525}{8192}a^{34}-\frac{35}{65536}a^{33}+\frac{32725}{32768}a^{32}+\frac{35}{2048}a^{31}-\frac{5797}{512}a^{30}-\frac{5425}{16384}a^{29}+\frac{49445}{512}a^{28}+\frac{35525}{8192}a^{27}-\frac{40455}{64}a^{26}-\frac{166257}{4096}a^{25}+\frac{13147875}{4096}a^{24}+\frac{143325}{512}a^{23}-\frac{3251625}{256}a^{22}-\frac{1480049}{1024}a^{21}+\frac{10015005}{256}a^{20}+\frac{1438927}{256}a^{19}-\frac{2982525}{32}a^{18}-\frac{262871}{16}a^{17}+\frac{21729825}{128}a^{16}+\frac{285957}{8}a^{15}-\frac{928625}{4}a^{14}-\frac{1818635}{32}a^{13}+\frac{928625}{4}a^{12}+\frac{2051777}{32}a^{11}-163438a^{10}-\frac{784343}{16}a^{9}+\frac{1225785}{16}a^{8}+\frac{188653}{8}a^{7}-21945a^{6}-6265a^{5}+3325a^{4}+707a^{3}-200a^{2}-21a+2$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}+\frac{35}{2048}a^{31}-\frac{5425}{16384}a^{29}+\frac{35525}{8192}a^{27}-\frac{166257}{4096}a^{25}+\frac{143325}{512}a^{23}-\frac{1480049}{1024}a^{21}+\frac{1438927}{256}a^{19}-\frac{262871}{16}a^{17}+\frac{285957}{8}a^{15}-\frac{1818635}{32}a^{13}+\frac{1}{64}a^{12}+\frac{2051777}{32}a^{11}-\frac{3}{8}a^{10}-\frac{784343}{16}a^{9}+\frac{27}{8}a^{8}+\frac{188653}{8}a^{7}-14a^{6}-6265a^{5}+\frac{105}{4}a^{4}+707a^{3}-18a^{2}-21a+2$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}+\frac{35}{2048}a^{31}-\frac{5425}{16384}a^{29}+\frac{35525}{8192}a^{27}-\frac{166257}{4096}a^{25}+\frac{143325}{512}a^{23}-\frac{1480049}{1024}a^{21}+\frac{1438927}{256}a^{19}-\frac{262871}{16}a^{17}+\frac{1}{256}a^{16}+\frac{285957}{8}a^{15}-\frac{1}{8}a^{14}-\frac{1818635}{32}a^{13}+\frac{13}{8}a^{12}+\frac{2051777}{32}a^{11}-11a^{10}-\frac{784343}{16}a^{9}+\frac{165}{4}a^{8}+\frac{188653}{8}a^{7}-84a^{6}-6265a^{5}+84a^{4}+707a^{3}-32a^{2}-21a+2$, $\frac{1}{1048576}a^{40}-\frac{5}{65536}a^{38}+\frac{185}{65536}a^{36}-\frac{1}{131072}a^{35}-\frac{525}{8192}a^{34}+\frac{35}{65536}a^{33}+\frac{32725}{32768}a^{32}-\frac{35}{2048}a^{31}-\frac{371007}{32768}a^{30}+\frac{5425}{16384}a^{29}+\frac{791105}{8192}a^{28}-\frac{35525}{8192}a^{27}-\frac{2588917}{4096}a^{26}+\frac{166257}{4096}a^{25}+\frac{13144599}{4096}a^{24}-\frac{143325}{512}a^{23}-\frac{25995451}{2048}a^{22}+\frac{1480049}{1024}a^{21}+\frac{19997131}{512}a^{20}-\frac{1438927}{256}a^{19}-\frac{23771755}{256}a^{18}+\frac{262871}{16}a^{17}+\frac{43114687}{256}a^{16}-\frac{285957}{8}a^{15}-\frac{14614789}{64}a^{14}+\frac{1818635}{32}a^{13}+\frac{14369253}{64}a^{12}-\frac{2051777}{32}a^{11}-\frac{611255}{4}a^{10}+\frac{784343}{16}a^{9}+\frac{1069979}{16}a^{8}-\frac{188653}{8}a^{7}-\frac{132391}{8}a^{6}+6265a^{5}+\frac{7175}{4}a^{4}-707a^{3}-\frac{105}{2}a^{2}+21a$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}+\frac{35}{2048}a^{31}-\frac{1}{32768}a^{30}-\frac{5425}{16384}a^{29}+\frac{15}{8192}a^{28}+\frac{35525}{8192}a^{27}-\frac{405}{8192}a^{26}-\frac{166257}{4096}a^{25}+\frac{1625}{2048}a^{24}+\frac{143325}{512}a^{23}-\frac{8625}{1024}a^{22}-\frac{1480049}{1024}a^{21}+\frac{15939}{256}a^{20}+\frac{1438927}{256}a^{19}-\frac{168245}{512}a^{18}-\frac{262871}{16}a^{17}+\frac{159885}{128}a^{16}+\frac{285957}{8}a^{15}-\frac{218025}{64}a^{14}-\frac{1818635}{32}a^{13}+\frac{104975}{16}a^{12}+\frac{2051777}{32}a^{11}-\frac{138567}{16}a^{10}-\frac{784343}{16}a^{9}+\frac{29835}{4}a^{8}+\frac{188653}{8}a^{7}-\frac{7735}{2}a^{6}-6265a^{5}+1050a^{4}+707a^{3}-\frac{225}{2}a^{2}-21a+2$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}+\frac{35}{2048}a^{31}-\frac{5425}{16384}a^{29}+\frac{35525}{8192}a^{27}+\frac{1}{8192}a^{26}-\frac{166257}{4096}a^{25}-\frac{13}{2048}a^{24}+\frac{143325}{512}a^{23}+\frac{299}{2048}a^{22}-\frac{1480049}{1024}a^{21}-\frac{1001}{512}a^{20}+\frac{1438927}{256}a^{19}+\frac{8645}{512}a^{18}-\frac{262871}{16}a^{17}-\frac{12597}{128}a^{16}+\frac{285957}{8}a^{15}+\frac{12597}{32}a^{14}-\frac{1818635}{32}a^{13}-\frac{8619}{8}a^{12}+\frac{2051777}{32}a^{11}+\frac{31603}{16}a^{10}-\frac{784343}{16}a^{9}-\frac{9295}{4}a^{8}+\frac{188653}{8}a^{7}+\frac{13013}{8}a^{6}-6265a^{5}-\frac{1183}{2}a^{4}+707a^{3}+\frac{169}{2}a^{2}-21a-2$, $\frac{1}{131072}a^{35}-\frac{35}{65536}a^{33}+\frac{35}{2048}a^{31}-\frac{5425}{16384}a^{29}+\frac{35525}{8192}a^{27}-\frac{166257}{4096}a^{25}+\frac{143325}{512}a^{23}-\frac{1480049}{1024}a^{21}+\frac{1438927}{256}a^{19}-\frac{262871}{16}a^{17}+\frac{285957}{8}a^{15}-\frac{1818635}{32}a^{13}+\frac{2051777}{32}a^{11}-\frac{784343}{16}a^{9}+\frac{188653}{8}a^{7}-6265a^{5}+707a^{3}-21a+1$
| 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: | \( 31575205777918616000000000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{42}\cdot(2\pi)^{0}\cdot 31575205777918616000000000000 \cdot 1}{2\cdot\sqrt{155718699466313184257207094263668545441599708733396657696588937331033553383727300608}}\cr\approx \mathstrut & 0.175956626859625 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^42 - 84*x^40 + 3276*x^38 - 78736*x^36 + 1305360*x^34 - 15833664*x^32 + 145435136*x^30 - 1032886144*x^28 + 5741625344*x^26 - 25131545600*x^24 + 86701812736*x^22 - 234915702784*x^20 + 495818960896*x^18 - 804378279936*x^16 + 983365402624*x^14 - 880319397888*x^12 + 553414557696*x^10 - 229269569536*x^8 + 56479711232*x^6 - 6910640128*x^4 + 308281344*x^2 - 2097152)
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^42 - 84*x^40 + 3276*x^38 - 78736*x^36 + 1305360*x^34 - 15833664*x^32 + 145435136*x^30 - 1032886144*x^28 + 5741625344*x^26 - 25131545600*x^24 + 86701812736*x^22 - 234915702784*x^20 + 495818960896*x^18 - 804378279936*x^16 + 983365402624*x^14 - 880319397888*x^12 + 553414557696*x^10 - 229269569536*x^8 + 56479711232*x^6 - 6910640128*x^4 + 308281344*x^2 - 2097152, 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^42 - 84*x^40 + 3276*x^38 - 78736*x^36 + 1305360*x^34 - 15833664*x^32 + 145435136*x^30 - 1032886144*x^28 + 5741625344*x^26 - 25131545600*x^24 + 86701812736*x^22 - 234915702784*x^20 + 495818960896*x^18 - 804378279936*x^16 + 983365402624*x^14 - 880319397888*x^12 + 553414557696*x^10 - 229269569536*x^8 + 56479711232*x^6 - 6910640128*x^4 + 308281344*x^2 - 2097152);
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^42 - 84*x^40 + 3276*x^38 - 78736*x^36 + 1305360*x^34 - 15833664*x^32 + 145435136*x^30 - 1032886144*x^28 + 5741625344*x^26 - 25131545600*x^24 + 86701812736*x^22 - 234915702784*x^20 + 495818960896*x^18 - 804378279936*x^16 + 983365402624*x^14 - 880319397888*x^12 + 553414557696*x^10 - 229269569536*x^8 + 56479711232*x^6 - 6910640128*x^4 + 308281344*x^2 - 2097152);
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
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 42 |
| The 42 conjugacy class representatives for $C_{42}$ |
| Character table for $C_{42}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), 6.6.1229312.1, 7.7.13841287201.1, 14.14.401774962552217617093885952.1, \(\Q(\zeta_{49})^+\) |
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 | $42$ | $42$ | R | $42$ | ${\href{/padicField/13.14.0.1}{14} }^{3}$ | $21^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{7}$ | $21^{2}$ | ${\href{/padicField/29.14.0.1}{14} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{14}$ | $42$ | ${\href{/padicField/41.7.0.1}{7} }^{6}$ | ${\href{/padicField/43.14.0.1}{14} }^{3}$ | $21^{2}$ | $42$ | $42$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $42$ | $2$ | $21$ | $63$ | |||
|
\(7\)
| Deg $21$ | $21$ | $1$ | $38$ | |||
| Deg $21$ | $21$ | $1$ | $38$ |