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Properties

Label 40.0.204...561.1
Degree $40$
Signature $[0, 20]$
Discriminant $2.048\times 10^{69}$
Root discriminant \(54.05\)
Ramified primes $3,11,13$
Class number not computed
Class group not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - 3*x^38 + 10*x^37 - 10*x^36 - 30*x^35 + 127*x^34 - 127*x^33 - 381*x^32 + 1540*x^31 - 1540*x^30 + 5017*x^29 + 9192*x^28 - 47740*x^27 + 39883*x^26 + 133500*x^25 - 518980*x^24 + 534289*x^23 + 1583184*x^22 - 6478780*x^21 + 6419731*x^20 + 19436340*x^19 + 14248656*x^18 - 14425803*x^17 - 42037380*x^16 - 32440500*x^15 + 29074707*x^14 + 104407380*x^13 + 60308712*x^12 - 98749611*x^11 - 90935460*x^10 - 272806380*x^9 - 202479021*x^8 + 202479021*x^7 + 607437063*x^6 + 430467210*x^5 - 430467210*x^4 - 1291401630*x^3 - 1162261467*x^2 + 1162261467*x + 3486784401)
 
gp: K = bnfinit(y^40 - y^39 - 3*y^38 + 10*y^37 - 10*y^36 - 30*y^35 + 127*y^34 - 127*y^33 - 381*y^32 + 1540*y^31 - 1540*y^30 + 5017*y^29 + 9192*y^28 - 47740*y^27 + 39883*y^26 + 133500*y^25 - 518980*y^24 + 534289*y^23 + 1583184*y^22 - 6478780*y^21 + 6419731*y^20 + 19436340*y^19 + 14248656*y^18 - 14425803*y^17 - 42037380*y^16 - 32440500*y^15 + 29074707*y^14 + 104407380*y^13 + 60308712*y^12 - 98749611*y^11 - 90935460*y^10 - 272806380*y^9 - 202479021*y^8 + 202479021*y^7 + 607437063*y^6 + 430467210*y^5 - 430467210*y^4 - 1291401630*y^3 - 1162261467*y^2 + 1162261467*y + 3486784401, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - x^39 - 3*x^38 + 10*x^37 - 10*x^36 - 30*x^35 + 127*x^34 - 127*x^33 - 381*x^32 + 1540*x^31 - 1540*x^30 + 5017*x^29 + 9192*x^28 - 47740*x^27 + 39883*x^26 + 133500*x^25 - 518980*x^24 + 534289*x^23 + 1583184*x^22 - 6478780*x^21 + 6419731*x^20 + 19436340*x^19 + 14248656*x^18 - 14425803*x^17 - 42037380*x^16 - 32440500*x^15 + 29074707*x^14 + 104407380*x^13 + 60308712*x^12 - 98749611*x^11 - 90935460*x^10 - 272806380*x^9 - 202479021*x^8 + 202479021*x^7 + 607437063*x^6 + 430467210*x^5 - 430467210*x^4 - 1291401630*x^3 - 1162261467*x^2 + 1162261467*x + 3486784401);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - x^39 - 3*x^38 + 10*x^37 - 10*x^36 - 30*x^35 + 127*x^34 - 127*x^33 - 381*x^32 + 1540*x^31 - 1540*x^30 + 5017*x^29 + 9192*x^28 - 47740*x^27 + 39883*x^26 + 133500*x^25 - 518980*x^24 + 534289*x^23 + 1583184*x^22 - 6478780*x^21 + 6419731*x^20 + 19436340*x^19 + 14248656*x^18 - 14425803*x^17 - 42037380*x^16 - 32440500*x^15 + 29074707*x^14 + 104407380*x^13 + 60308712*x^12 - 98749611*x^11 - 90935460*x^10 - 272806380*x^9 - 202479021*x^8 + 202479021*x^7 + 607437063*x^6 + 430467210*x^5 - 430467210*x^4 - 1291401630*x^3 - 1162261467*x^2 + 1162261467*x + 3486784401)
 

\( x^{40} - x^{39} - 3 x^{38} + 10 x^{37} - 10 x^{36} - 30 x^{35} + 127 x^{34} - 127 x^{33} + \cdots + 3486784401 \) Copy content Toggle raw display

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:   \(2048466217933115502043842255668249817786415413279145255704960577548561\) \(\medspace = 3^{20}\cdot 11^{36}\cdot 13^{20}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(54.05\)
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:  $3^{1/2}11^{9/10}13^{1/2}\approx 54.04875818839011$
Ramified primes:   \(3\), \(11\), \(13\) Copy content Toggle raw display
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:  \(429=3\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{429}(1,·)$, $\chi_{429}(131,·)$, $\chi_{429}(389,·)$, $\chi_{429}(391,·)$, $\chi_{429}(142,·)$, $\chi_{429}(272,·)$, $\chi_{429}(274,·)$, $\chi_{429}(259,·)$, $\chi_{429}(404,·)$, $\chi_{429}(25,·)$, $\chi_{429}(155,·)$, $\chi_{429}(157,·)$, $\chi_{429}(415,·)$, $\chi_{429}(38,·)$, $\chi_{429}(40,·)$, $\chi_{429}(170,·)$, $\chi_{429}(428,·)$, $\chi_{429}(53,·)$, $\chi_{429}(311,·)$, $\chi_{429}(313,·)$, $\chi_{429}(287,·)$, $\chi_{429}(181,·)$, $\chi_{429}(64,·)$, $\chi_{429}(194,·)$, $\chi_{429}(196,·)$, $\chi_{429}(326,·)$, $\chi_{429}(79,·)$, $\chi_{429}(337,·)$, $\chi_{429}(14,·)$, $\chi_{429}(248,·)$, $\chi_{429}(92,·)$, $\chi_{429}(350,·)$, $\chi_{429}(103,·)$, $\chi_{429}(233,·)$, $\chi_{429}(235,·)$, $\chi_{429}(365,·)$, $\chi_{429}(116,·)$, $\chi_{429}(118,·)$, $\chi_{429}(376,·)$, $\chi_{429}(298,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3}a^{21}-\frac{1}{3}a^{20}+\frac{1}{3}a^{18}-\frac{1}{3}a^{17}+\frac{1}{3}a^{15}-\frac{1}{3}a^{14}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{24147}a^{22}-\frac{1}{9}a^{21}-\frac{1}{3}a^{20}+\frac{1}{9}a^{19}-\frac{1}{9}a^{18}-\frac{1}{3}a^{17}+\frac{1}{9}a^{16}-\frac{1}{9}a^{15}-\frac{1}{3}a^{14}+\frac{1}{9}a^{13}-\frac{1}{9}a^{12}+\frac{6160}{24147}a^{11}+\frac{1}{3}a^{10}-\frac{4}{9}a^{9}+\frac{4}{9}a^{8}+\frac{1}{3}a^{7}-\frac{4}{9}a^{6}+\frac{4}{9}a^{5}+\frac{1}{3}a^{4}-\frac{4}{9}a^{3}+\frac{4}{9}a^{2}+\frac{1}{3}a-\frac{902}{2683}$, $\frac{1}{72441}a^{23}-\frac{1}{72441}a^{22}-\frac{1}{9}a^{21}+\frac{1}{27}a^{20}-\frac{1}{27}a^{19}-\frac{1}{9}a^{18}+\frac{10}{27}a^{17}-\frac{10}{27}a^{16}-\frac{1}{9}a^{15}-\frac{8}{27}a^{14}+\frac{8}{27}a^{13}-\frac{17987}{72441}a^{12}+\frac{4207}{24147}a^{11}+\frac{5}{27}a^{10}-\frac{5}{27}a^{9}+\frac{4}{9}a^{8}-\frac{4}{27}a^{7}+\frac{4}{27}a^{6}+\frac{4}{9}a^{5}-\frac{13}{27}a^{4}+\frac{13}{27}a^{3}+\frac{4}{9}a^{2}-\frac{1195}{2683}a+\frac{1195}{2683}$, $\frac{1}{217323}a^{24}-\frac{1}{217323}a^{23}-\frac{1}{72441}a^{22}+\frac{1}{81}a^{21}-\frac{1}{81}a^{20}-\frac{1}{27}a^{19}+\frac{37}{81}a^{18}-\frac{37}{81}a^{17}-\frac{10}{27}a^{16}-\frac{8}{81}a^{15}+\frac{8}{81}a^{14}-\frac{17987}{217323}a^{13}-\frac{19940}{72441}a^{12}-\frac{45310}{217323}a^{11}-\frac{5}{81}a^{10}-\frac{5}{27}a^{9}+\frac{23}{81}a^{8}-\frac{23}{81}a^{7}+\frac{4}{27}a^{6}-\frac{40}{81}a^{5}+\frac{40}{81}a^{4}+\frac{13}{27}a^{3}-\frac{3878}{8049}a^{2}+\frac{3878}{8049}a+\frac{1195}{2683}$, $\frac{1}{651969}a^{25}-\frac{1}{651969}a^{24}-\frac{1}{217323}a^{23}+\frac{10}{651969}a^{22}-\frac{1}{243}a^{21}-\frac{1}{81}a^{20}+\frac{37}{243}a^{19}-\frac{37}{243}a^{18}-\frac{37}{81}a^{17}-\frac{89}{243}a^{16}+\frac{89}{243}a^{15}-\frac{235310}{651969}a^{14}-\frac{19940}{217323}a^{13}-\frac{45310}{651969}a^{12}+\frac{37453}{651969}a^{11}-\frac{5}{81}a^{10}+\frac{104}{243}a^{9}-\frac{104}{243}a^{8}-\frac{23}{81}a^{7}-\frac{40}{243}a^{6}+\frac{40}{243}a^{5}+\frac{40}{81}a^{4}+\frac{4171}{24147}a^{3}-\frac{4171}{24147}a^{2}+\frac{3878}{8049}a+\frac{759}{2683}$, $\frac{1}{1955907}a^{26}-\frac{1}{1955907}a^{25}-\frac{1}{651969}a^{24}+\frac{10}{1955907}a^{23}-\frac{10}{1955907}a^{22}-\frac{1}{243}a^{21}+\frac{37}{729}a^{20}-\frac{37}{729}a^{19}-\frac{37}{243}a^{18}-\frac{332}{729}a^{17}+\frac{332}{729}a^{16}-\frac{887279}{1955907}a^{15}-\frac{237263}{651969}a^{14}-\frac{697279}{1955907}a^{13}+\frac{689422}{1955907}a^{12}+\frac{42070}{651969}a^{11}+\frac{347}{729}a^{10}-\frac{347}{729}a^{9}-\frac{104}{243}a^{8}-\frac{40}{729}a^{7}+\frac{40}{729}a^{6}+\frac{40}{243}a^{5}+\frac{4171}{72441}a^{4}-\frac{4171}{72441}a^{3}-\frac{4171}{24147}a^{2}+\frac{253}{2683}a-\frac{253}{2683}$, $\frac{1}{5867721}a^{27}-\frac{1}{5867721}a^{26}-\frac{1}{1955907}a^{25}+\frac{10}{5867721}a^{24}-\frac{10}{5867721}a^{23}-\frac{10}{1955907}a^{22}+\frac{37}{2187}a^{21}-\frac{37}{2187}a^{20}-\frac{37}{729}a^{19}+\frac{397}{2187}a^{18}-\frac{397}{2187}a^{17}-\frac{887279}{5867721}a^{16}-\frac{237263}{1955907}a^{15}-\frac{2653186}{5867721}a^{14}+\frac{689422}{5867721}a^{13}-\frac{609899}{1955907}a^{12}-\frac{2481448}{5867721}a^{11}+\frac{382}{2187}a^{10}-\frac{347}{729}a^{9}+\frac{689}{2187}a^{8}-\frac{689}{2187}a^{7}+\frac{40}{729}a^{6}+\frac{76612}{217323}a^{5}-\frac{76612}{217323}a^{4}-\frac{4171}{72441}a^{3}+\frac{253}{8049}a^{2}-\frac{253}{8049}a-\frac{253}{2683}$, $\frac{1}{17603163}a^{28}-\frac{1}{17603163}a^{27}-\frac{1}{5867721}a^{26}+\frac{10}{17603163}a^{25}-\frac{10}{17603163}a^{24}-\frac{10}{5867721}a^{23}+\frac{127}{17603163}a^{22}+\frac{692}{6561}a^{21}+\frac{692}{2187}a^{20}-\frac{332}{6561}a^{19}+\frac{332}{6561}a^{18}-\frac{887279}{17603163}a^{17}-\frac{889232}{5867721}a^{16}-\frac{697279}{17603163}a^{15}+\frac{689422}{17603163}a^{14}+\frac{694039}{5867721}a^{13}-\frac{525541}{17603163}a^{12}+\frac{540850}{17603163}a^{11}-\frac{347}{2187}a^{10}-\frac{769}{6561}a^{9}+\frac{769}{6561}a^{8}+\frac{769}{2187}a^{7}-\frac{285593}{651969}a^{6}+\frac{285593}{651969}a^{5}+\frac{68270}{217323}a^{4}+\frac{10985}{24147}a^{3}-\frac{10985}{24147}a^{2}-\frac{2936}{8049}a-\frac{746}{2683}$, $\frac{1}{52809489}a^{29}-\frac{1}{52809489}a^{28}-\frac{1}{17603163}a^{27}+\frac{10}{52809489}a^{26}-\frac{10}{52809489}a^{25}-\frac{10}{17603163}a^{24}+\frac{127}{52809489}a^{23}-\frac{127}{52809489}a^{22}+\frac{692}{6561}a^{21}+\frac{6229}{19683}a^{20}-\frac{6229}{19683}a^{19}+\frac{16715884}{52809489}a^{18}-\frac{889232}{17603163}a^{17}-\frac{697279}{52809489}a^{16}+\frac{689422}{52809489}a^{15}+\frac{694039}{17603163}a^{14}+\frac{17077622}{52809489}a^{13}-\frac{17062313}{52809489}a^{12}+\frac{534289}{17603163}a^{11}-\frac{7330}{19683}a^{10}+\frac{7330}{19683}a^{9}+\frac{769}{6561}a^{8}-\frac{937562}{1955907}a^{7}+\frac{937562}{1955907}a^{6}+\frac{285593}{651969}a^{5}+\frac{10985}{72441}a^{4}-\frac{10985}{72441}a^{3}-\frac{10985}{24147}a^{2}-\frac{1143}{2683}a+\frac{1143}{2683}$, $\frac{1}{633713868}a^{30}-\frac{1}{158428467}a^{29}+\frac{19}{633713868}a^{27}-\frac{10}{158428467}a^{26}+\frac{217}{633713868}a^{24}-\frac{127}{158428467}a^{23}+\frac{1}{236196}a^{21}-\frac{6229}{59049}a^{20}+\frac{56320393}{158428467}a^{19}-\frac{217}{23470884}a^{18}+\frac{15443}{59049}a^{17}-\frac{38911802}{158428467}a^{16}+\frac{19}{869292}a^{15}-\frac{13753}{59049}a^{14}+\frac{22533322}{158428467}a^{13}-\frac{1}{32196}a^{12}-\frac{15814}{59049}a^{11}-\frac{12353}{59049}a^{10}+\frac{1}{4}a^{9}-\frac{5619769}{17603163}a^{8}+\frac{937562}{5867721}a^{7}-\frac{1}{4}a^{6}+\frac{262825}{651969}a^{5}-\frac{10985}{217323}a^{4}+\frac{1}{4}a^{3}+\frac{9938}{24147}a^{2}-\frac{1540}{8049}a-\frac{1}{4}$, $\frac{1}{44\!\cdots\!16}a^{31}+\frac{28594415}{44\!\cdots\!16}a^{30}-\frac{14013983}{36\!\cdots\!43}a^{29}+\frac{53790139}{44\!\cdots\!16}a^{28}+\frac{704664281}{44\!\cdots\!16}a^{27}-\frac{140139830}{36\!\cdots\!43}a^{26}+\frac{537901417}{44\!\cdots\!16}a^{25}+\frac{7818692015}{44\!\cdots\!16}a^{24}-\frac{1779775841}{36\!\cdots\!43}a^{23}+\frac{6831347923}{44\!\cdots\!16}a^{22}-\frac{821161166209}{16483336322652}a^{21}-\frac{51\!\cdots\!48}{11\!\cdots\!29}a^{20}+\frac{18\!\cdots\!59}{14\!\cdots\!72}a^{19}-\frac{42\!\cdots\!17}{44\!\cdots\!16}a^{18}-\frac{38\!\cdots\!82}{11\!\cdots\!29}a^{17}+\frac{43\!\cdots\!25}{14\!\cdots\!72}a^{16}-\frac{13\!\cdots\!07}{44\!\cdots\!16}a^{15}-\frac{352084612416752}{11\!\cdots\!29}a^{14}+\frac{49\!\cdots\!11}{14\!\cdots\!72}a^{13}+\frac{94\!\cdots\!67}{44\!\cdots\!16}a^{12}+\frac{21\!\cdots\!56}{11\!\cdots\!29}a^{11}+\frac{1966534567799}{5494445440884}a^{10}-\frac{12877790160661}{16\!\cdots\!08}a^{9}-\frac{157773269818457}{409488808830327}a^{8}+\frac{152320795521853}{545985078440436}a^{7}+\frac{1054621677247}{60665008715604}a^{6}-\frac{3312898030378}{15166252178901}a^{5}+\frac{9170342938445}{20221669571868}a^{4}-\frac{86354569933}{2246852174652}a^{3}+\frac{231815859388}{561713043663}a^{2}+\frac{50525557181}{748950724884}a+\frac{21230664213}{249650241628}$, $\frac{1}{13\!\cdots\!48}a^{32}-\frac{1}{13\!\cdots\!48}a^{31}+\frac{29238911}{44\!\cdots\!16}a^{30}+\frac{1044866827}{13\!\cdots\!48}a^{29}-\frac{1395733771}{13\!\cdots\!48}a^{28}-\frac{840194443}{44\!\cdots\!16}a^{27}+\frac{10448668297}{13\!\cdots\!48}a^{26}-\frac{13957337737}{13\!\cdots\!48}a^{25}-\frac{7612493833}{44\!\cdots\!16}a^{24}+\frac{132698087299}{13\!\cdots\!48}a^{23}-\frac{177258189187}{13\!\cdots\!48}a^{22}+\frac{94\!\cdots\!43}{13\!\cdots\!48}a^{21}-\frac{14\!\cdots\!77}{44\!\cdots\!16}a^{20}+\frac{18\!\cdots\!39}{13\!\cdots\!48}a^{19}-\frac{60\!\cdots\!35}{13\!\cdots\!48}a^{18}+\frac{10\!\cdots\!65}{44\!\cdots\!16}a^{17}+\frac{14\!\cdots\!01}{13\!\cdots\!48}a^{16}+\frac{45\!\cdots\!55}{13\!\cdots\!48}a^{15}+\frac{21\!\cdots\!83}{44\!\cdots\!16}a^{14}-\frac{84\!\cdots\!29}{13\!\cdots\!48}a^{13}+\frac{13\!\cdots\!37}{13\!\cdots\!48}a^{12}+\frac{20\!\cdots\!09}{44\!\cdots\!16}a^{11}-\frac{629368259146375}{16\!\cdots\!08}a^{10}+\frac{334149918188345}{49\!\cdots\!24}a^{9}-\frac{805342369068079}{16\!\cdots\!08}a^{8}-\frac{13393036025843}{60665008715604}a^{7}-\frac{27365079161339}{181995026146812}a^{6}-\frac{3251408266847}{60665008715604}a^{5}+\frac{938759033261}{2246852174652}a^{4}+\frac{2240708391665}{6740556523956}a^{3}+\frac{424210010441}{2246852174652}a^{2}+\frac{162752885357}{748950724884}a+\frac{66020788121}{249650241628}$, $\frac{1}{39\!\cdots\!44}a^{33}-\frac{1}{39\!\cdots\!44}a^{32}-\frac{1}{13\!\cdots\!48}a^{31}-\frac{67970270}{99\!\cdots\!61}a^{30}+\frac{2419271837}{39\!\cdots\!44}a^{29}-\frac{443915839}{13\!\cdots\!48}a^{28}-\frac{2290245761}{99\!\cdots\!61}a^{27}+\frac{24192718343}{39\!\cdots\!44}a^{26}-\frac{4439158417}{13\!\cdots\!48}a^{25}-\frac{24737654900}{99\!\cdots\!61}a^{24}+\frac{307247523029}{39\!\cdots\!44}a^{23}-\frac{7566706855301}{39\!\cdots\!44}a^{22}+\frac{23\!\cdots\!21}{33\!\cdots\!87}a^{21}-\frac{18\!\cdots\!77}{39\!\cdots\!44}a^{20}+\frac{12\!\cdots\!45}{39\!\cdots\!44}a^{19}+\frac{14\!\cdots\!51}{33\!\cdots\!87}a^{18}-\frac{18\!\cdots\!03}{39\!\cdots\!44}a^{17}+\frac{254636639091283}{39\!\cdots\!44}a^{16}+\frac{13\!\cdots\!55}{33\!\cdots\!87}a^{15}+\frac{75\!\cdots\!55}{39\!\cdots\!44}a^{14}+\frac{11\!\cdots\!93}{39\!\cdots\!44}a^{13}+\frac{44\!\cdots\!35}{33\!\cdots\!87}a^{12}+\frac{10\!\cdots\!85}{49\!\cdots\!24}a^{11}+\frac{625913220433787}{49\!\cdots\!24}a^{10}+\frac{56034022321492}{136496269610109}a^{9}+\frac{203454476552419}{545985078440436}a^{8}+\frac{46143438988555}{181995026146812}a^{7}+\frac{1724231306756}{5055417392967}a^{6}+\frac{1375581142703}{20221669571868}a^{5}-\frac{171205428205}{6740556523956}a^{4}+\frac{277832431946}{561713043663}a^{3}-\frac{580897479011}{2246852174652}a^{2}-\frac{305834551243}{748950724884}a-\frac{32355177521}{249650241628}$, $\frac{1}{11\!\cdots\!32}a^{34}-\frac{1}{11\!\cdots\!32}a^{33}-\frac{1}{39\!\cdots\!44}a^{32}+\frac{5}{59\!\cdots\!66}a^{31}+\frac{844237637}{11\!\cdots\!32}a^{30}+\frac{702866717}{39\!\cdots\!44}a^{29}-\frac{2742775321}{59\!\cdots\!66}a^{28}-\frac{416137141}{11\!\cdots\!32}a^{27}+\frac{7028667143}{39\!\cdots\!44}a^{26}-\frac{27427753075}{59\!\cdots\!66}a^{25}+\frac{18633044789}{11\!\cdots\!32}a^{24}-\frac{7129782701465}{11\!\cdots\!32}a^{23}+\frac{1116818331833}{19\!\cdots\!22}a^{22}+\frac{13\!\cdots\!15}{11\!\cdots\!32}a^{21}+\frac{15\!\cdots\!65}{11\!\cdots\!32}a^{20}-\frac{55\!\cdots\!45}{19\!\cdots\!22}a^{19}+\frac{26\!\cdots\!21}{11\!\cdots\!32}a^{18}+\frac{37\!\cdots\!19}{11\!\cdots\!32}a^{17}-\frac{39\!\cdots\!67}{19\!\cdots\!22}a^{16}+\frac{15\!\cdots\!55}{11\!\cdots\!32}a^{15}-\frac{43\!\cdots\!79}{11\!\cdots\!32}a^{14}+\frac{85\!\cdots\!35}{19\!\cdots\!22}a^{13}+\frac{22\!\cdots\!75}{44\!\cdots\!16}a^{12}-\frac{10\!\cdots\!47}{44\!\cdots\!16}a^{11}-\frac{34\!\cdots\!51}{73\!\cdots\!86}a^{10}-\frac{300183611489561}{16\!\cdots\!08}a^{9}+\frac{140599207547341}{16\!\cdots\!08}a^{8}-\frac{71468685694219}{272992539220218}a^{7}-\frac{27013604348869}{60665008715604}a^{6}+\frac{24229636820921}{60665008715604}a^{5}-\frac{1781235400673}{10110834785934}a^{4}-\frac{2703841837535}{6740556523956}a^{3}+\frac{700204572701}{2246852174652}a^{2}+\frac{305758738313}{748950724884}a-\frac{17017037980}{62412560407}$, $\frac{1}{35\!\cdots\!96}a^{35}-\frac{1}{35\!\cdots\!96}a^{34}-\frac{1}{11\!\cdots\!32}a^{33}+\frac{5}{17\!\cdots\!98}a^{32}-\frac{5}{17\!\cdots\!98}a^{31}-\frac{31582753}{29\!\cdots\!83}a^{30}+\frac{1455853037}{17\!\cdots\!98}a^{29}-\frac{697867025}{17\!\cdots\!98}a^{28}-\frac{949005772}{29\!\cdots\!83}a^{27}+\frac{14558530505}{17\!\cdots\!98}a^{26}-\frac{6978670385}{17\!\cdots\!98}a^{25}-\frac{1880422106111}{89\!\cdots\!49}a^{24}+\frac{1294560265655}{59\!\cdots\!66}a^{23}+\frac{11007733266223}{17\!\cdots\!98}a^{22}-\frac{49\!\cdots\!86}{89\!\cdots\!49}a^{21}+\frac{10\!\cdots\!71}{59\!\cdots\!66}a^{20}-\frac{40\!\cdots\!89}{17\!\cdots\!98}a^{19}+\frac{18\!\cdots\!66}{89\!\cdots\!49}a^{18}-\frac{14\!\cdots\!29}{59\!\cdots\!66}a^{17}-\frac{48\!\cdots\!27}{17\!\cdots\!98}a^{16}-\frac{97\!\cdots\!60}{89\!\cdots\!49}a^{15}-\frac{21\!\cdots\!41}{59\!\cdots\!66}a^{14}-\frac{15\!\cdots\!83}{66\!\cdots\!74}a^{13}-\frac{65\!\cdots\!94}{33\!\cdots\!87}a^{12}+\frac{11\!\cdots\!13}{22\!\cdots\!58}a^{11}-\frac{30\!\cdots\!35}{73\!\cdots\!86}a^{10}-\frac{3191652703546}{15166252178901}a^{9}+\frac{386618897325247}{818977617660654}a^{8}-\frac{122371332986525}{272992539220218}a^{7}-\frac{649113203183}{1685139130989}a^{6}+\frac{14816178895595}{30332504357802}a^{5}+\frac{4639694472889}{10110834785934}a^{4}+\frac{785301167422}{1685139130989}a^{3}-\frac{651342412183}{2246852174652}a^{2}+\frac{304335814685}{748950724884}a+\frac{5320253099}{249650241628}$, $\frac{1}{10\!\cdots\!88}a^{36}-\frac{1}{10\!\cdots\!88}a^{35}-\frac{1}{35\!\cdots\!96}a^{34}+\frac{5}{53\!\cdots\!94}a^{33}-\frac{5}{53\!\cdots\!94}a^{32}-\frac{5}{17\!\cdots\!98}a^{31}+\frac{1204537849}{26\!\cdots\!47}a^{30}-\frac{36893895089}{53\!\cdots\!94}a^{29}+\frac{9085864099}{17\!\cdots\!98}a^{28}+\frac{63772607644}{26\!\cdots\!47}a^{27}-\frac{368938951025}{53\!\cdots\!94}a^{26}-\frac{3426211537351}{53\!\cdots\!94}a^{25}+\frac{839880776107}{89\!\cdots\!49}a^{24}+\frac{6410837702095}{53\!\cdots\!94}a^{23}-\frac{33526160381491}{53\!\cdots\!94}a^{22}-\frac{19\!\cdots\!06}{89\!\cdots\!49}a^{21}-\frac{20\!\cdots\!61}{53\!\cdots\!94}a^{20}-\frac{23\!\cdots\!29}{53\!\cdots\!94}a^{19}-\frac{15\!\cdots\!86}{89\!\cdots\!49}a^{18}-\frac{12\!\cdots\!21}{53\!\cdots\!94}a^{17}-\frac{95\!\cdots\!35}{53\!\cdots\!94}a^{16}-\frac{28\!\cdots\!24}{89\!\cdots\!49}a^{15}-\frac{54\!\cdots\!47}{19\!\cdots\!22}a^{14}+\frac{38\!\cdots\!05}{19\!\cdots\!22}a^{13}-\frac{14\!\cdots\!22}{33\!\cdots\!87}a^{12}-\frac{24\!\cdots\!75}{73\!\cdots\!86}a^{11}+\frac{34\!\cdots\!75}{73\!\cdots\!86}a^{10}+\frac{489272950510900}{12\!\cdots\!81}a^{9}-\frac{58907080786085}{272992539220218}a^{8}+\frac{75160167566635}{272992539220218}a^{7}-\frac{21744159197914}{45498756536703}a^{6}+\frac{4391093071331}{30332504357802}a^{5}-\frac{4702906785869}{10110834785934}a^{4}+\frac{6585793915}{6740556523956}a^{3}-\frac{909929684839}{2246852174652}a^{2}-\frac{74201510653}{249650241628}a+\frac{13267702943}{124825120814}$, $\frac{1}{32\!\cdots\!64}a^{37}-\frac{1}{32\!\cdots\!64}a^{36}-\frac{1}{10\!\cdots\!88}a^{35}+\frac{5}{16\!\cdots\!82}a^{34}-\frac{5}{16\!\cdots\!82}a^{33}-\frac{5}{53\!\cdots\!94}a^{32}+\frac{127}{32\!\cdots\!64}a^{31}+\frac{12982906673}{32\!\cdots\!64}a^{30}+\frac{245257771}{53\!\cdots\!94}a^{29}-\frac{53403172667}{32\!\cdots\!64}a^{28}+\frac{86465705039}{32\!\cdots\!64}a^{27}-\frac{3691429727191}{16\!\cdots\!82}a^{26}+\frac{2287847732197}{10\!\cdots\!88}a^{25}+\frac{23407920290057}{32\!\cdots\!64}a^{24}-\frac{36894431392459}{16\!\cdots\!82}a^{23}+\frac{22397848767967}{10\!\cdots\!88}a^{22}-\frac{19\!\cdots\!37}{32\!\cdots\!64}a^{21}-\frac{57\!\cdots\!21}{16\!\cdots\!82}a^{20}+\frac{51\!\cdots\!05}{10\!\cdots\!88}a^{19}-\frac{78\!\cdots\!75}{32\!\cdots\!64}a^{18}-\frac{36\!\cdots\!25}{16\!\cdots\!82}a^{17}-\frac{12\!\cdots\!37}{10\!\cdots\!88}a^{16}+\frac{40\!\cdots\!33}{11\!\cdots\!32}a^{15}-\frac{20\!\cdots\!25}{59\!\cdots\!66}a^{14}-\frac{14\!\cdots\!81}{39\!\cdots\!44}a^{13}-\frac{80\!\cdots\!87}{44\!\cdots\!16}a^{12}-\frac{10\!\cdots\!03}{22\!\cdots\!58}a^{11}+\frac{885404075758255}{14\!\cdots\!72}a^{10}+\frac{13\!\cdots\!95}{49\!\cdots\!24}a^{9}-\frac{136483524497893}{272992539220218}a^{8}-\frac{236312237816833}{545985078440436}a^{7}-\frac{15583109022239}{60665008715604}a^{6}-\frac{1967699686711}{30332504357802}a^{5}+\frac{3056656890755}{10110834785934}a^{4}+\frac{92028532453}{1123426087326}a^{3}-\frac{390834509863}{2246852174652}a^{2}+\frac{84147236359}{249650241628}a-\frac{29047258363}{249650241628}$, $\frac{1}{96\!\cdots\!92}a^{38}-\frac{1}{96\!\cdots\!92}a^{37}-\frac{1}{32\!\cdots\!64}a^{36}+\frac{5}{48\!\cdots\!46}a^{35}-\frac{5}{48\!\cdots\!46}a^{34}-\frac{5}{16\!\cdots\!82}a^{33}+\frac{127}{96\!\cdots\!92}a^{32}-\frac{127}{96\!\cdots\!92}a^{31}-\frac{589525018}{80\!\cdots\!41}a^{30}-\frac{869717885039}{96\!\cdots\!92}a^{29}+\frac{898015084379}{96\!\cdots\!92}a^{28}-\frac{1209485340890}{24\!\cdots\!23}a^{27}-\frac{433201309043}{32\!\cdots\!64}a^{26}+\frac{31172875599857}{96\!\cdots\!92}a^{25}-\frac{12142604935358}{24\!\cdots\!23}a^{24}-\frac{12159474055781}{32\!\cdots\!64}a^{23}+\frac{335975163276803}{96\!\cdots\!92}a^{22}-\frac{19\!\cdots\!28}{24\!\cdots\!23}a^{21}-\frac{71\!\cdots\!87}{32\!\cdots\!64}a^{20}-\frac{25\!\cdots\!03}{96\!\cdots\!92}a^{19}-\frac{67\!\cdots\!11}{24\!\cdots\!23}a^{18}+\frac{11\!\cdots\!63}{32\!\cdots\!64}a^{17}-\frac{88\!\cdots\!31}{35\!\cdots\!96}a^{16}+\frac{25\!\cdots\!97}{89\!\cdots\!49}a^{15}+\frac{37\!\cdots\!07}{11\!\cdots\!32}a^{14}+\frac{63\!\cdots\!61}{13\!\cdots\!48}a^{13}+\frac{12\!\cdots\!52}{33\!\cdots\!87}a^{12}+\frac{24\!\cdots\!71}{44\!\cdots\!16}a^{11}-\frac{14\!\cdots\!25}{14\!\cdots\!72}a^{10}+\frac{568730880144643}{12\!\cdots\!81}a^{9}-\frac{190607819908715}{545985078440436}a^{8}+\frac{2653704538427}{60665008715604}a^{7}-\frac{5315621434798}{15166252178901}a^{6}-\frac{3729260979941}{10110834785934}a^{5}+\frac{49433310785}{1123426087326}a^{4}-\frac{31034567675}{2246852174652}a^{3}-\frac{85605002713}{748950724884}a^{2}-\frac{283777865059}{748950724884}a+\frac{49760573921}{124825120814}$, $\frac{1}{29\!\cdots\!76}a^{39}-\frac{1}{29\!\cdots\!76}a^{38}-\frac{1}{96\!\cdots\!92}a^{37}+\frac{5}{14\!\cdots\!38}a^{36}-\frac{5}{14\!\cdots\!38}a^{35}-\frac{5}{48\!\cdots\!46}a^{34}+\frac{127}{29\!\cdots\!76}a^{33}-\frac{127}{29\!\cdots\!76}a^{32}-\frac{127}{96\!\cdots\!92}a^{31}+\frac{44126511082}{72\!\cdots\!69}a^{30}-\frac{2029210757461}{29\!\cdots\!76}a^{29}-\frac{6074388339683}{29\!\cdots\!76}a^{28}+\frac{1226729125495}{24\!\cdots\!23}a^{27}+\frac{1900617181457}{29\!\cdots\!76}a^{26}-\frac{60743883407117}{29\!\cdots\!76}a^{25}+\frac{12664429854688}{24\!\cdots\!23}a^{24}-\frac{35782518636877}{29\!\cdots\!76}a^{23}-\frac{771447319242611}{29\!\cdots\!76}a^{22}+\frac{30\!\cdots\!93}{24\!\cdots\!23}a^{21}-\frac{85\!\cdots\!35}{29\!\cdots\!76}a^{20}+\frac{43\!\cdots\!27}{29\!\cdots\!76}a^{19}+\frac{93\!\cdots\!32}{24\!\cdots\!23}a^{18}-\frac{52\!\cdots\!47}{10\!\cdots\!88}a^{17}-\frac{43\!\cdots\!49}{10\!\cdots\!88}a^{16}-\frac{44\!\cdots\!29}{89\!\cdots\!49}a^{15}+\frac{65\!\cdots\!57}{39\!\cdots\!44}a^{14}+\frac{10\!\cdots\!07}{39\!\cdots\!44}a^{13}+\frac{16\!\cdots\!11}{33\!\cdots\!87}a^{12}+\frac{22\!\cdots\!71}{49\!\cdots\!24}a^{11}-\frac{107632952496419}{545985078440436}a^{10}-\frac{140877745336174}{12\!\cdots\!81}a^{9}-\frac{198334185757103}{545985078440436}a^{8}+\frac{266974510112491}{545985078440436}a^{7}-\frac{8957514151145}{181995026146812}a^{6}+\frac{360242276995}{10110834785934}a^{5}-\frac{1036242651755}{10110834785934}a^{4}+\frac{1386925587613}{3370278261978}a^{3}-\frac{112769309511}{249650241628}a^{2}+\frac{132948970925}{748950724884}a-\frac{53358221139}{249650241628}$ 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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:   \( \frac{59541067}{132674374061025948} a^{34} + \frac{6160}{1425856203} a^{23} + \frac{59541067}{1425856203} a^{12} - \frac{573797262679}{748950724884} a \)  (order $66$) Copy content Toggle raw display
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 - x^39 - 3*x^38 + 10*x^37 - 10*x^36 - 30*x^35 + 127*x^34 - 127*x^33 - 381*x^32 + 1540*x^31 - 1540*x^30 + 5017*x^29 + 9192*x^28 - 47740*x^27 + 39883*x^26 + 133500*x^25 - 518980*x^24 + 534289*x^23 + 1583184*x^22 - 6478780*x^21 + 6419731*x^20 + 19436340*x^19 + 14248656*x^18 - 14425803*x^17 - 42037380*x^16 - 32440500*x^15 + 29074707*x^14 + 104407380*x^13 + 60308712*x^12 - 98749611*x^11 - 90935460*x^10 - 272806380*x^9 - 202479021*x^8 + 202479021*x^7 + 607437063*x^6 + 430467210*x^5 - 430467210*x^4 - 1291401630*x^3 - 1162261467*x^2 + 1162261467*x + 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 - x^39 - 3*x^38 + 10*x^37 - 10*x^36 - 30*x^35 + 127*x^34 - 127*x^33 - 381*x^32 + 1540*x^31 - 1540*x^30 + 5017*x^29 + 9192*x^28 - 47740*x^27 + 39883*x^26 + 133500*x^25 - 518980*x^24 + 534289*x^23 + 1583184*x^22 - 6478780*x^21 + 6419731*x^20 + 19436340*x^19 + 14248656*x^18 - 14425803*x^17 - 42037380*x^16 - 32440500*x^15 + 29074707*x^14 + 104407380*x^13 + 60308712*x^12 - 98749611*x^11 - 90935460*x^10 - 272806380*x^9 - 202479021*x^8 + 202479021*x^7 + 607437063*x^6 + 430467210*x^5 - 430467210*x^4 - 1291401630*x^3 - 1162261467*x^2 + 1162261467*x + 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 - x^39 - 3*x^38 + 10*x^37 - 10*x^36 - 30*x^35 + 127*x^34 - 127*x^33 - 381*x^32 + 1540*x^31 - 1540*x^30 + 5017*x^29 + 9192*x^28 - 47740*x^27 + 39883*x^26 + 133500*x^25 - 518980*x^24 + 534289*x^23 + 1583184*x^22 - 6478780*x^21 + 6419731*x^20 + 19436340*x^19 + 14248656*x^18 - 14425803*x^17 - 42037380*x^16 - 32440500*x^15 + 29074707*x^14 + 104407380*x^13 + 60308712*x^12 - 98749611*x^11 - 90935460*x^10 - 272806380*x^9 - 202479021*x^8 + 202479021*x^7 + 607437063*x^6 + 430467210*x^5 - 430467210*x^4 - 1291401630*x^3 - 1162261467*x^2 + 1162261467*x + 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 - x^39 - 3*x^38 + 10*x^37 - 10*x^36 - 30*x^35 + 127*x^34 - 127*x^33 - 381*x^32 + 1540*x^31 - 1540*x^30 + 5017*x^29 + 9192*x^28 - 47740*x^27 + 39883*x^26 + 133500*x^25 - 518980*x^24 + 534289*x^23 + 1583184*x^22 - 6478780*x^21 + 6419731*x^20 + 19436340*x^19 + 14248656*x^18 - 14425803*x^17 - 42037380*x^16 - 32440500*x^15 + 29074707*x^14 + 104407380*x^13 + 60308712*x^12 - 98749611*x^11 - 90935460*x^10 - 272806380*x^9 - 202479021*x^8 + 202479021*x^7 + 607437063*x^6 + 430467210*x^5 - 430467210*x^4 - 1291401630*x^3 - 1162261467*x^2 + 1162261467*x + 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

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{429}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-143}) \), \(\Q(\sqrt{13}, \sqrt{33})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{33}, \sqrt{-39})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-11}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{-143})\), \(\Q(\sqrt{-11}, \sqrt{-39})\), \(\Q(\zeta_{11})^+\), 8.0.33871089681.2, \(\Q(\zeta_{33})^+\), 10.10.79589952003133.1, 10.10.212743941704374509.1, 10.0.52089208083.1, \(\Q(\zeta_{11})\), 10.0.19340358336761319.3, 10.0.875489472034463.1, 20.20.45259984731914299465343386928991081.1, \(\Q(\zeta_{33})\), 20.0.45259984731914299465343386928991081.3, 20.0.374049460594333053432589974619761.1, 20.0.766481815643182771348259698369.1, 20.0.45259984731914299465343386928991081.4, 20.0.45259984731914299465343386928991081.5

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 ${\href{/padicField/2.10.0.1}{10} }^{4}$ R ${\href{/padicField/5.10.0.1}{10} }^{4}$ ${\href{/padicField/7.10.0.1}{10} }^{4}$ R R ${\href{/padicField/17.10.0.1}{10} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{20}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{20}$ ${\href{/padicField/47.10.0.1}{10} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{4}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.10.5.2$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(11\) Copy content Toggle raw display 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}$
\(13\) Copy content Toggle raw display 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$$10$20T3$[\ ]_{2}^{10}$
13.20.10.1$x^{20} + 1300 x^{19} + 760630 x^{18} + 263792100 x^{17} + 60057199605 x^{16} + 9380582749214 x^{15} + 1018311780763300 x^{14} + 75907334238787106 x^{13} + 3723649138838156452 x^{12} + 108997457660124507008 x^{11} + 1475055936539742904023 x^{10} + 1416974086834508372240 x^{9} + 629829636823684464902 x^{8} + 192953738203144224036 x^{7} + 798096976259371806456 x^{6} + 10514315760388324731550 x^{5} + 12773049284749524150248 x^{4} + 15044084853570106847092 x^{3} + 5322693419045842540420 x^{2} + 2225999864943544940488 x + 2898376971411614164801$$2$$10$$10$20T3$[\ ]_{2}^{10}$