Normalized defining polynomial
\( x^{40} + 3 x^{38} + 5 x^{36} + 3 x^{34} - 11 x^{32} - 45 x^{30} - 91 x^{28} - 93 x^{26} + 85 x^{24} + \cdots + 1048576 \)
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: | \(2712047505327472012144594086234943329183650768987920496114727387136\) \(\medspace = 2^{40}\cdot 7^{20}\cdot 11^{36}\) | 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: | \(2\), \(7\), \(11\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{6164}a^{22}-\frac{1}{4}a^{20}+\frac{1}{4}a^{18}-\frac{1}{4}a^{16}+\frac{1}{4}a^{14}-\frac{1}{4}a^{12}+\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}+\frac{627}{1541}$, $\frac{1}{12328}a^{23}-\frac{1}{8}a^{21}+\frac{1}{8}a^{19}-\frac{1}{8}a^{17}+\frac{1}{8}a^{15}-\frac{1}{8}a^{13}+\frac{1}{8}a^{11}-\frac{1}{8}a^{9}+\frac{1}{8}a^{7}-\frac{1}{8}a^{5}+\frac{1}{8}a^{3}+\frac{627}{3082}a$, $\frac{1}{24656}a^{24}-\frac{1}{24656}a^{22}+\frac{5}{16}a^{20}+\frac{3}{16}a^{18}+\frac{5}{16}a^{16}+\frac{3}{16}a^{14}+\frac{5}{16}a^{12}+\frac{3}{16}a^{10}+\frac{5}{16}a^{8}+\frac{3}{16}a^{6}+\frac{5}{16}a^{4}-\frac{457}{3082}a^{2}-\frac{542}{1541}$, $\frac{1}{49312}a^{25}-\frac{1}{49312}a^{23}+\frac{5}{32}a^{21}+\frac{3}{32}a^{19}-\frac{11}{32}a^{17}-\frac{13}{32}a^{15}+\frac{5}{32}a^{13}+\frac{3}{32}a^{11}-\frac{11}{32}a^{9}-\frac{13}{32}a^{7}+\frac{5}{32}a^{5}-\frac{457}{6164}a^{3}-\frac{271}{1541}a$, $\frac{1}{98624}a^{26}-\frac{1}{98624}a^{24}-\frac{7}{98624}a^{22}-\frac{29}{64}a^{20}+\frac{21}{64}a^{18}-\frac{13}{64}a^{16}+\frac{5}{64}a^{14}+\frac{3}{64}a^{12}-\frac{11}{64}a^{10}+\frac{19}{64}a^{8}-\frac{27}{64}a^{6}-\frac{457}{12328}a^{4}-\frac{271}{3082}a^{2}-\frac{178}{1541}$, $\frac{1}{197248}a^{27}-\frac{1}{197248}a^{25}-\frac{7}{197248}a^{23}-\frac{29}{128}a^{21}+\frac{21}{128}a^{19}+\frac{51}{128}a^{17}-\frac{59}{128}a^{15}+\frac{3}{128}a^{13}-\frac{11}{128}a^{11}-\frac{45}{128}a^{9}+\frac{37}{128}a^{7}-\frac{457}{24656}a^{5}-\frac{271}{6164}a^{3}-\frac{89}{1541}a$, $\frac{1}{394496}a^{28}-\frac{1}{394496}a^{26}-\frac{7}{394496}a^{24}-\frac{17}{394496}a^{22}-\frac{107}{256}a^{20}+\frac{51}{256}a^{18}+\frac{69}{256}a^{16}+\frac{3}{256}a^{14}-\frac{11}{256}a^{12}-\frac{45}{256}a^{10}-\frac{91}{256}a^{8}-\frac{457}{49312}a^{6}-\frac{271}{12328}a^{4}-\frac{89}{3082}a^{2}+\frac{2}{1541}$, $\frac{1}{788992}a^{29}-\frac{1}{788992}a^{27}-\frac{7}{788992}a^{25}-\frac{17}{788992}a^{23}-\frac{107}{512}a^{21}-\frac{205}{512}a^{19}-\frac{187}{512}a^{17}-\frac{253}{512}a^{15}-\frac{11}{512}a^{13}-\frac{45}{512}a^{11}-\frac{91}{512}a^{9}-\frac{457}{98624}a^{7}-\frac{271}{24656}a^{5}-\frac{89}{6164}a^{3}+\frac{1}{1541}a$, $\frac{1}{1577984}a^{30}-\frac{1}{1577984}a^{28}-\frac{7}{1577984}a^{26}-\frac{17}{1577984}a^{24}-\frac{1}{68608}a^{22}+\frac{307}{1024}a^{20}+\frac{325}{1024}a^{18}-\frac{253}{1024}a^{16}-\frac{11}{1024}a^{14}-\frac{45}{1024}a^{12}-\frac{91}{1024}a^{10}-\frac{457}{197248}a^{8}-\frac{271}{49312}a^{6}-\frac{89}{12328}a^{4}+\frac{1}{3082}a^{2}+\frac{2}{67}$, $\frac{1}{3155968}a^{31}-\frac{1}{3155968}a^{29}-\frac{7}{3155968}a^{27}-\frac{17}{3155968}a^{25}-\frac{1}{137216}a^{23}+\frac{307}{2048}a^{21}-\frac{699}{2048}a^{19}+\frac{771}{2048}a^{17}+\frac{1013}{2048}a^{15}-\frac{45}{2048}a^{13}-\frac{91}{2048}a^{11}-\frac{457}{394496}a^{9}-\frac{271}{98624}a^{7}-\frac{89}{24656}a^{5}+\frac{1}{6164}a^{3}+\frac{1}{67}a$, $\frac{1}{6311936}a^{32}-\frac{1}{6311936}a^{30}-\frac{7}{6311936}a^{28}-\frac{17}{6311936}a^{26}-\frac{1}{274432}a^{24}-\frac{1}{6311936}a^{22}-\frac{699}{4096}a^{20}+\frac{771}{4096}a^{18}+\frac{1013}{4096}a^{16}-\frac{45}{4096}a^{14}-\frac{91}{4096}a^{12}-\frac{457}{788992}a^{10}-\frac{271}{197248}a^{8}-\frac{89}{49312}a^{6}+\frac{1}{12328}a^{4}+\frac{1}{134}a^{2}+\frac{34}{1541}$, $\frac{1}{12623872}a^{33}-\frac{1}{12623872}a^{31}-\frac{7}{12623872}a^{29}-\frac{17}{12623872}a^{27}-\frac{1}{548864}a^{25}-\frac{1}{12623872}a^{23}-\frac{699}{8192}a^{21}-\frac{3325}{8192}a^{19}+\frac{1013}{8192}a^{17}-\frac{45}{8192}a^{15}+\frac{4005}{8192}a^{13}+\frac{788535}{1577984}a^{11}+\frac{196977}{394496}a^{9}+\frac{49223}{98624}a^{7}-\frac{12327}{24656}a^{5}-\frac{133}{268}a^{3}-\frac{1507}{3082}a$, $\frac{1}{25247744}a^{34}-\frac{1}{25247744}a^{32}-\frac{7}{25247744}a^{30}-\frac{17}{25247744}a^{28}-\frac{1}{1097728}a^{26}-\frac{1}{25247744}a^{24}+\frac{89}{25247744}a^{22}-\frac{7421}{16384}a^{20}-\frac{3083}{16384}a^{18}+\frac{4051}{16384}a^{16}+\frac{8101}{16384}a^{14}-\frac{457}{3155968}a^{12}-\frac{271}{788992}a^{10}-\frac{89}{197248}a^{8}+\frac{1}{49312}a^{6}+\frac{1}{536}a^{4}+\frac{17}{3082}a^{2}+\frac{14}{1541}$, $\frac{1}{50495488}a^{35}-\frac{1}{50495488}a^{33}-\frac{7}{50495488}a^{31}-\frac{17}{50495488}a^{29}-\frac{1}{2195456}a^{27}-\frac{1}{50495488}a^{25}+\frac{89}{50495488}a^{23}-\frac{7421}{32768}a^{21}-\frac{3083}{32768}a^{19}-\frac{12333}{32768}a^{17}+\frac{8101}{32768}a^{15}-\frac{457}{6311936}a^{13}-\frac{271}{1577984}a^{11}-\frac{89}{394496}a^{9}+\frac{1}{98624}a^{7}+\frac{1}{1072}a^{5}+\frac{17}{6164}a^{3}+\frac{7}{1541}a$, $\frac{1}{100990976}a^{36}-\frac{1}{100990976}a^{34}-\frac{7}{100990976}a^{32}-\frac{17}{100990976}a^{30}-\frac{1}{4390912}a^{28}-\frac{1}{100990976}a^{26}+\frac{89}{100990976}a^{24}+\frac{271}{100990976}a^{22}+\frac{29685}{65536}a^{20}-\frac{12333}{65536}a^{18}-\frac{24667}{65536}a^{16}-\frac{457}{12623872}a^{14}-\frac{271}{3155968}a^{12}-\frac{89}{788992}a^{10}+\frac{1}{197248}a^{8}+\frac{1}{2144}a^{6}+\frac{17}{12328}a^{4}+\frac{7}{3082}a^{2}+\frac{2}{1541}$, $\frac{1}{201981952}a^{37}-\frac{1}{201981952}a^{35}-\frac{7}{201981952}a^{33}-\frac{17}{201981952}a^{31}-\frac{1}{8781824}a^{29}-\frac{1}{201981952}a^{27}+\frac{89}{201981952}a^{25}+\frac{271}{201981952}a^{23}+\frac{29685}{131072}a^{21}-\frac{12333}{131072}a^{19}-\frac{24667}{131072}a^{17}-\frac{457}{25247744}a^{15}-\frac{271}{6311936}a^{13}-\frac{89}{1577984}a^{11}+\frac{1}{394496}a^{9}+\frac{1}{4288}a^{7}+\frac{17}{24656}a^{5}+\frac{7}{6164}a^{3}+\frac{1}{1541}a$, $\frac{1}{403963904}a^{38}-\frac{1}{403963904}a^{36}-\frac{7}{403963904}a^{34}-\frac{17}{403963904}a^{32}-\frac{1}{17563648}a^{30}-\frac{1}{403963904}a^{28}+\frac{89}{403963904}a^{26}+\frac{271}{403963904}a^{24}+\frac{457}{403963904}a^{22}+\frac{118739}{262144}a^{20}-\frac{24667}{262144}a^{18}-\frac{457}{50495488}a^{16}-\frac{271}{12623872}a^{14}-\frac{89}{3155968}a^{12}+\frac{1}{788992}a^{10}+\frac{1}{8576}a^{8}+\frac{17}{49312}a^{6}+\frac{7}{12328}a^{4}+\frac{1}{3082}a^{2}-\frac{2}{1541}$, $\frac{1}{807927808}a^{39}-\frac{1}{807927808}a^{37}-\frac{7}{807927808}a^{35}-\frac{17}{807927808}a^{33}-\frac{1}{35127296}a^{31}-\frac{1}{807927808}a^{29}+\frac{89}{807927808}a^{27}+\frac{271}{807927808}a^{25}+\frac{457}{807927808}a^{23}+\frac{118739}{524288}a^{21}+\frac{237477}{524288}a^{19}-\frac{457}{100990976}a^{17}-\frac{271}{25247744}a^{15}-\frac{89}{6311936}a^{13}+\frac{1}{1577984}a^{11}+\frac{1}{17152}a^{9}+\frac{17}{98624}a^{7}+\frac{7}{24656}a^{5}+\frac{1}{6164}a^{3}-\frac{1}{1541}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{1}{49312} a^{27} + \frac{8279}{49312} a^{5} \) (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 $
Galois group
$C_2^2\times C_{10}$ (as 40T7):
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
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | Deg $20$ | $2$ | $10$ | $20$ | |||
Deg $20$ | $2$ | $10$ | $20$ | ||||
\(7\) | 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 | $[\ ]_{2}^{10}$ |
7.20.10.1 | $x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ | |
\(11\) | 11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |
11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |