Normalized defining polynomial
\( x^{20} - 30 x^{18} - 80 x^{17} + 555 x^{16} + 1920 x^{15} - 1000 x^{14} + 40560 x^{13} + \cdots - 122509757583 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(4, 8)$ |
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| Discriminant: |
\(1498476178648776262633390080000000000000000000000\)
\(\medspace = 2^{51}\cdot 3^{20}\cdot 5^{22}\cdot 23^{2}\cdot 389^{2}\)
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| |
| Root discriminant: | \(256.32\) |
| |
| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(23\), \(389\)
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| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{10}+\frac{1}{8}a^{8}-\frac{3}{8}a^{6}-\frac{1}{8}a^{4}+\frac{1}{8}a^{2}+\frac{1}{8}$, $\frac{1}{8}a^{11}+\frac{1}{8}a^{9}-\frac{3}{8}a^{7}-\frac{1}{8}a^{5}+\frac{1}{8}a^{3}+\frac{1}{8}a$, $\frac{1}{8}a^{12}-\frac{1}{2}a^{8}+\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{8}$, $\frac{1}{8}a^{13}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{8}a^{14}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}+\frac{3}{8}a^{2}-\frac{1}{2}$, $\frac{1}{24}a^{15}-\frac{1}{24}a^{14}+\frac{1}{24}a^{13}+\frac{1}{12}a^{9}+\frac{5}{12}a^{8}+\frac{1}{3}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{11}{24}a^{3}-\frac{11}{24}a^{2}+\frac{11}{24}a-\frac{1}{2}$, $\frac{1}{72}a^{16}-\frac{1}{36}a^{13}+\frac{1}{24}a^{11}-\frac{1}{72}a^{10}+\frac{5}{24}a^{9}+\frac{1}{24}a^{8}-\frac{13}{72}a^{7}+\frac{11}{24}a^{6}+\frac{7}{24}a^{5}-\frac{5}{36}a^{4}+\frac{1}{24}a^{3}-\frac{1}{24}a^{2}+\frac{17}{72}a+\frac{7}{24}$, $\frac{1}{648}a^{17}-\frac{1}{216}a^{15}+\frac{7}{162}a^{14}-\frac{1}{54}a^{13}+\frac{1}{216}a^{12}-\frac{1}{648}a^{11}-\frac{7}{216}a^{10}+\frac{23}{216}a^{9}+\frac{47}{648}a^{8}-\frac{17}{72}a^{7}+\frac{7}{216}a^{6}+\frac{139}{324}a^{5}+\frac{49}{216}a^{4}-\frac{1}{6}a^{3}-\frac{49}{648}a^{2}+\frac{11}{216}a-\frac{2}{9}$, $\frac{1}{11664}a^{18}+\frac{1}{1944}a^{17}-\frac{1}{3888}a^{16}-\frac{19}{1458}a^{15}-\frac{55}{1944}a^{14}-\frac{25}{1944}a^{13}-\frac{97}{2916}a^{12}+\frac{1}{54}a^{11}-\frac{181}{3888}a^{10}+\frac{95}{2916}a^{9}+\frac{637}{3888}a^{8}-\frac{169}{486}a^{7}+\frac{329}{729}a^{6}+\frac{721}{1944}a^{5}+\frac{79}{648}a^{4}-\frac{721}{2916}a^{3}-\frac{29}{1296}a^{2}-\frac{11}{216}a-\frac{95}{432}$, $\frac{1}{73\cdots 48}a^{19}-\frac{72\cdots 51}{24\cdots 16}a^{18}-\frac{45\cdots 99}{24\cdots 16}a^{17}-\frac{32\cdots 31}{73\cdots 48}a^{16}-\frac{14\cdots 13}{13\cdots 12}a^{15}+\frac{23\cdots 27}{12\cdots 08}a^{14}-\frac{40\cdots 27}{36\cdots 24}a^{13}+\frac{57\cdots 31}{12\cdots 08}a^{12}-\frac{26\cdots 59}{24\cdots 16}a^{11}+\frac{29\cdots 05}{73\cdots 48}a^{10}-\frac{47\cdots 29}{24\cdots 16}a^{9}+\frac{16\cdots 45}{24\cdots 16}a^{8}+\frac{16\cdots 91}{36\cdots 24}a^{7}+\frac{37\cdots 51}{13\cdots 12}a^{6}-\frac{59\cdots 45}{15\cdots 68}a^{5}+\frac{31\cdots 41}{36\cdots 24}a^{4}+\frac{36\cdots 11}{24\cdots 16}a^{3}-\frac{25\cdots 83}{27\cdots 24}a^{2}-\frac{59\cdots 71}{27\cdots 24}a-\frac{29\cdots 87}{90\cdots 08}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ (assuming GRH) |
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Unit group
| Rank: | $11$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{10\cdots 58}{85\cdots 07}a^{19}+\frac{37\cdots 41}{10\cdots 84}a^{18}-\frac{22\cdots 24}{85\cdots 07}a^{17}-\frac{50\cdots 45}{28\cdots 69}a^{16}+\frac{44\cdots 93}{25\cdots 21}a^{15}+\frac{49\cdots 15}{17\cdots 14}a^{14}+\frac{22\cdots 68}{31\cdots 41}a^{13}+\frac{73\cdots 85}{10\cdots 84}a^{12}-\frac{28\cdots 12}{28\cdots 69}a^{11}+\frac{78\cdots 37}{17\cdots 14}a^{10}+\frac{50\cdots 72}{25\cdots 21}a^{9}-\frac{48\cdots 87}{85\cdots 07}a^{8}-\frac{20\cdots 98}{28\cdots 69}a^{7}+\frac{60\cdots 57}{10\cdots 84}a^{6}-\frac{13\cdots 80}{85\cdots 07}a^{5}+\frac{13\cdots 41}{17\cdots 14}a^{4}-\frac{10\cdots 43}{25\cdots 21}a^{3}+\frac{55\cdots 73}{56\cdots 38}a^{2}-\frac{10\cdots 38}{94\cdots 23}a+\frac{19\cdots 31}{37\cdots 92}$, $\frac{21\cdots 37}{22\cdots 89}a^{19}+\frac{40\cdots 75}{15\cdots 26}a^{18}-\frac{15\cdots 06}{76\cdots 63}a^{17}-\frac{12\cdots 55}{91\cdots 56}a^{16}+\frac{33\cdots 13}{25\cdots 21}a^{15}+\frac{65\cdots 67}{30\cdots 52}a^{14}+\frac{12\cdots 68}{22\cdots 89}a^{13}+\frac{16\cdots 39}{30\cdots 52}a^{12}-\frac{56\cdots 40}{76\cdots 63}a^{11}+\frac{15\cdots 49}{45\cdots 78}a^{10}+\frac{11\cdots 32}{76\cdots 63}a^{9}-\frac{32\cdots 89}{76\cdots 63}a^{8}-\frac{12\cdots 79}{22\cdots 89}a^{7}+\frac{22\cdots 35}{51\cdots 42}a^{6}-\frac{96\cdots 40}{85\cdots 07}a^{5}+\frac{55\cdots 89}{91\cdots 56}a^{4}-\frac{22\cdots 85}{76\cdots 63}a^{3}+\frac{24\cdots 49}{34\cdots 28}a^{2}-\frac{72\cdots 82}{85\cdots 07}a+\frac{44\cdots 19}{11\cdots 76}$, $\frac{22\cdots 21}{73\cdots 48}a^{19}+\frac{18\cdots 57}{24\cdots 16}a^{18}-\frac{18\cdots 47}{24\cdots 16}a^{17}-\frac{32\cdots 51}{73\cdots 48}a^{16}+\frac{23\cdots 47}{40\cdots 36}a^{15}+\frac{92\cdots 27}{12\cdots 08}a^{14}+\frac{63\cdots 99}{36\cdots 24}a^{13}+\frac{20\cdots 89}{12\cdots 08}a^{12}-\frac{61\cdots 59}{24\cdots 16}a^{11}+\frac{11\cdots 29}{73\cdots 48}a^{10}+\frac{12\cdots 91}{24\cdots 16}a^{9}-\frac{37\cdots 47}{24\cdots 16}a^{8}-\frac{61\cdots 35}{36\cdots 24}a^{7}+\frac{62\cdots 85}{40\cdots 36}a^{6}-\frac{54\cdots 75}{13\cdots 12}a^{5}+\frac{75\cdots 97}{36\cdots 24}a^{4}-\frac{24\cdots 61}{24\cdots 16}a^{3}+\frac{69\cdots 37}{27\cdots 24}a^{2}-\frac{85\cdots 67}{27\cdots 24}a+\frac{13\cdots 29}{90\cdots 08}$, $\frac{22\cdots 51}{61\cdots 04}a^{19}+\frac{26\cdots 97}{40\cdots 36}a^{18}-\frac{49\cdots 85}{51\cdots 42}a^{17}-\frac{56\cdots 01}{12\cdots 08}a^{16}+\frac{11\cdots 57}{10\cdots 84}a^{15}+\frac{18\cdots 89}{20\cdots 68}a^{14}+\frac{39\cdots 79}{30\cdots 52}a^{13}+\frac{11\cdots 53}{68\cdots 56}a^{12}-\frac{32\cdots 27}{10\cdots 84}a^{11}+\frac{48\cdots 61}{12\cdots 08}a^{10}+\frac{61\cdots 57}{10\cdots 84}a^{9}-\frac{90\cdots 03}{40\cdots 36}a^{8}-\frac{82\cdots 99}{61\cdots 04}a^{7}+\frac{42\cdots 99}{20\cdots 68}a^{6}-\frac{19\cdots 75}{34\cdots 28}a^{5}+\frac{79\cdots 53}{30\cdots 52}a^{4}-\frac{50\cdots 23}{37\cdots 92}a^{3}+\frac{18\cdots 49}{50\cdots 56}a^{2}-\frac{54\cdots 71}{11\cdots 76}a+\frac{12\cdots 99}{50\cdots 56}$, $\frac{12\cdots 33}{36\cdots 24}a^{19}+\frac{47\cdots 65}{30\cdots 52}a^{18}-\frac{59\cdots 41}{12\cdots 08}a^{17}-\frac{26\cdots 87}{45\cdots 78}a^{16}-\frac{15\cdots 01}{22\cdots 52}a^{15}+\frac{87\cdots 81}{15\cdots 26}a^{14}+\frac{13\cdots 67}{45\cdots 78}a^{13}+\frac{83\cdots 15}{30\cdots 52}a^{12}-\frac{28\cdots 73}{12\cdots 08}a^{11}-\frac{37\cdots 29}{18\cdots 12}a^{10}+\frac{59\cdots 57}{12\cdots 08}a^{9}-\frac{54\cdots 73}{61\cdots 04}a^{8}-\frac{23\cdots 61}{91\cdots 56}a^{7}+\frac{32\cdots 53}{28\cdots 92}a^{6}-\frac{71\cdots 15}{22\cdots 52}a^{5}+\frac{36\cdots 43}{18\cdots 12}a^{4}-\frac{10\cdots 83}{12\cdots 08}a^{3}+\frac{11\cdots 77}{68\cdots 56}a^{2}-\frac{24\cdots 39}{13\cdots 12}a+\frac{16\cdots 63}{22\cdots 52}$, $\frac{34\cdots 37}{45\cdots 78}a^{19}+\frac{13\cdots 67}{61\cdots 04}a^{18}-\frac{10\cdots 75}{61\cdots 04}a^{17}-\frac{24\cdots 77}{22\cdots 89}a^{16}+\frac{37\cdots 25}{34\cdots 28}a^{15}+\frac{10\cdots 21}{61\cdots 04}a^{14}+\frac{79\cdots 07}{18\cdots 12}a^{13}+\frac{13\cdots 59}{30\cdots 52}a^{12}-\frac{37\cdots 71}{61\cdots 04}a^{11}+\frac{54\cdots 35}{18\cdots 12}a^{10}+\frac{73\cdots 09}{61\cdots 04}a^{9}-\frac{21\cdots 27}{61\cdots 04}a^{8}-\frac{81\cdots 13}{18\cdots 12}a^{7}+\frac{12\cdots 17}{34\cdots 28}a^{6}-\frac{10\cdots 85}{11\cdots 76}a^{5}+\frac{90\cdots 47}{18\cdots 12}a^{4}-\frac{14\cdots 31}{61\cdots 04}a^{3}+\frac{20\cdots 63}{34\cdots 28}a^{2}-\frac{24\cdots 27}{34\cdots 28}a+\frac{73\cdots 29}{22\cdots 52}$, $\frac{59\cdots 95}{36\cdots 24}a^{19}+\frac{25\cdots 35}{61\cdots 04}a^{18}-\frac{46\cdots 65}{12\cdots 08}a^{17}-\frac{41\cdots 51}{18\cdots 12}a^{16}+\frac{10\cdots 61}{34\cdots 28}a^{15}+\frac{23\cdots 99}{61\cdots 04}a^{14}+\frac{15\cdots 45}{18\cdots 12}a^{13}+\frac{52\cdots 67}{61\cdots 04}a^{12}-\frac{16\cdots 23}{12\cdots 08}a^{11}+\frac{84\cdots 79}{91\cdots 56}a^{10}+\frac{32\cdots 99}{12\cdots 08}a^{9}-\frac{61\cdots 43}{76\cdots 63}a^{8}-\frac{80\cdots 21}{91\cdots 56}a^{7}+\frac{55\cdots 79}{68\cdots 56}a^{6}-\frac{11\cdots 57}{56\cdots 38}a^{5}+\frac{19\cdots 01}{18\cdots 12}a^{4}-\frac{65\cdots 79}{12\cdots 08}a^{3}+\frac{92\cdots 23}{68\cdots 56}a^{2}-\frac{22\cdots 83}{13\cdots 12}a+\frac{17\cdots 77}{22\cdots 52}$, $\frac{32\cdots 27}{81\cdots 72}a^{19}+\frac{59\cdots 73}{81\cdots 72}a^{18}-\frac{28\cdots 41}{27\cdots 24}a^{17}-\frac{41\cdots 93}{81\cdots 72}a^{16}+\frac{51\cdots 71}{40\cdots 36}a^{15}+\frac{13\cdots 81}{13\cdots 12}a^{14}+\frac{57\cdots 27}{40\cdots 36}a^{13}+\frac{76\cdots 51}{40\cdots 36}a^{12}-\frac{94\cdots 01}{27\cdots 24}a^{11}+\frac{35\cdots 23}{81\cdots 72}a^{10}+\frac{54\cdots 75}{81\cdots 72}a^{9}-\frac{66\cdots 73}{27\cdots 24}a^{8}-\frac{61\cdots 73}{40\cdots 36}a^{7}+\frac{93\cdots 29}{40\cdots 36}a^{6}-\frac{84\cdots 45}{13\cdots 12}a^{5}+\frac{11\cdots 31}{40\cdots 36}a^{4}-\frac{12\cdots 77}{81\cdots 72}a^{3}+\frac{36\cdots 85}{90\cdots 08}a^{2}-\frac{53\cdots 15}{10\cdots 12}a+\frac{80\cdots 73}{30\cdots 36}$, $\frac{10\cdots 15}{73\cdots 48}a^{19}+\frac{67\cdots 23}{24\cdots 16}a^{18}-\frac{10\cdots 17}{24\cdots 16}a^{17}-\frac{16\cdots 01}{73\cdots 48}a^{16}+\frac{17\cdots 83}{40\cdots 36}a^{15}+\frac{53\cdots 47}{12\cdots 08}a^{14}+\frac{30\cdots 61}{36\cdots 24}a^{13}+\frac{85\cdots 69}{12\cdots 08}a^{12}-\frac{32\cdots 49}{24\cdots 16}a^{11}+\frac{74\cdots 83}{73\cdots 48}a^{10}+\frac{63\cdots 17}{24\cdots 16}a^{9}-\frac{19\cdots 05}{24\cdots 16}a^{8}-\frac{34\cdots 89}{36\cdots 24}a^{7}+\frac{32\cdots 35}{40\cdots 36}a^{6}-\frac{26\cdots 55}{13\cdots 12}a^{5}+\frac{37\cdots 11}{36\cdots 24}a^{4}-\frac{12\cdots 51}{24\cdots 16}a^{3}+\frac{34\cdots 03}{27\cdots 24}a^{2}-\frac{40\cdots 93}{27\cdots 24}a+\frac{60\cdots 31}{90\cdots 08}$, $\frac{41\cdots 01}{73\cdots 48}a^{19}+\frac{37\cdots 15}{24\cdots 16}a^{18}-\frac{36\cdots 63}{24\cdots 16}a^{17}-\frac{72\cdots 57}{73\cdots 48}a^{16}+\frac{19\cdots 45}{40\cdots 36}a^{15}+\frac{18\cdots 59}{12\cdots 08}a^{14}+\frac{16\cdots 63}{36\cdots 24}a^{13}+\frac{43\cdots 81}{12\cdots 08}a^{12}-\frac{11\cdots 31}{24\cdots 16}a^{11}+\frac{56\cdots 63}{73\cdots 48}a^{10}+\frac{23\cdots 59}{24\cdots 16}a^{9}-\frac{59\cdots 29}{24\cdots 16}a^{8}-\frac{15\cdots 33}{36\cdots 24}a^{7}+\frac{10\cdots 47}{40\cdots 36}a^{6}-\frac{88\cdots 79}{13\cdots 12}a^{5}+\frac{13\cdots 15}{36\cdots 24}a^{4}-\frac{43\cdots 17}{24\cdots 16}a^{3}+\frac{11\cdots 35}{27\cdots 24}a^{2}-\frac{12\cdots 23}{27\cdots 24}a+\frac{18\cdots 11}{90\cdots 08}$, $\frac{89\cdots 05}{73\cdots 48}a^{19}+\frac{84\cdots 39}{24\cdots 16}a^{18}-\frac{65\cdots 35}{24\cdots 16}a^{17}-\frac{12\cdots 65}{73\cdots 48}a^{16}+\frac{73\cdots 71}{40\cdots 36}a^{15}+\frac{34\cdots 93}{12\cdots 08}a^{14}+\frac{25\cdots 59}{36\cdots 24}a^{13}+\frac{84\cdots 17}{12\cdots 08}a^{12}-\frac{23\cdots 79}{24\cdots 16}a^{11}+\frac{35\cdots 75}{73\cdots 48}a^{10}+\frac{47\cdots 51}{24\cdots 16}a^{9}-\frac{13\cdots 21}{24\cdots 16}a^{8}-\frac{26\cdots 73}{36\cdots 24}a^{7}+\frac{23\cdots 57}{40\cdots 36}a^{6}-\frac{20\cdots 43}{13\cdots 12}a^{5}+\frac{29\cdots 07}{36\cdots 24}a^{4}-\frac{94\cdots 41}{24\cdots 16}a^{3}+\frac{26\cdots 91}{27\cdots 24}a^{2}-\frac{30\cdots 71}{27\cdots 24}a+\frac{47\cdots 59}{90\cdots 08}$
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| Regulator: | \( 72532444830100000 \) (assuming GRH) |
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| Unit signature rank: | \( 3 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 72532444830100000 \cdot 2}{2\cdot\sqrt{1498476178648776262633390080000000000000000000000}}\cr\approx \mathstrut & 2.30285357290926 \end{aligned}\] (assuming GRH)
Galois group
$C_2^8.(F_5\times S_4)$ (as 20T811):
| A solvable group of order 122880 |
| The 80 conjugacy class representatives for $C_2^8.(F_5\times S_4)$ |
| Character table for $C_2^8.(F_5\times S_4)$ |
Intermediate fields
| 5.1.200000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 30 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $20$ | R | $20$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }$ | $15{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 2.1.2.3a1.4 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ | |
| 2.2.8.48b6.64 | $x^{16} + 8 x^{15} + 40 x^{14} + 144 x^{13} + 406 x^{12} + 920 x^{11} + 1716 x^{10} + 2672 x^{9} + 3507 x^{8} + 3904 x^{7} + 3684 x^{6} + 2936 x^{5} + 1954 x^{4} + 1072 x^{3} + 476 x^{2} + 168 x + 39$ | $8$ | $2$ | $48$ | 16T939 | $$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{4}$$ | |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 3.1.3.4a1.1 | $x^{3} + 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $S_3$ | $$[2]^{2}$$ | |
| 3.4.1.0a1.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 3.4.3.16a1.1 | $x^{12} + 6 x^{11} + 12 x^{10} + 8 x^{9} + 9 x^{8} + 36 x^{7} + 36 x^{6} + 24 x^{4} + 48 x^{3} + 23$ | $3$ | $4$ | $16$ | $C_3 : C_4$ | $$[2]^{4}$$ | |
|
\(5\)
| 5.1.5.5a1.4 | $x^{5} + 20 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ |
| 5.1.15.17a1.4 | $x^{15} + 20 x^{3} + 5$ | $15$ | $1$ | $17$ | $F_5 \times S_3$ | $$[\frac{5}{4}]_{12}^{2}$$ | |
|
\(23\)
| 23.2.2.2a1.2 | $x^{4} + 42 x^{3} + 451 x^{2} + 210 x + 48$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 23.8.1.0a1.1 | $x^{8} + 3 x^{4} + 20 x^{3} + 5 x^{2} + 3 x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $$[\ ]^{8}$$ | |
| 23.8.1.0a1.1 | $x^{8} + 3 x^{4} + 20 x^{3} + 5 x^{2} + 3 x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $$[\ ]^{8}$$ | |
|
\(389\)
| $\Q_{389}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{389}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{389}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{389}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |