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Properties

Label 38.38.687...664.1
Degree $38$
Signature $[38, 0]$
Discriminant $6.877\times 10^{95}$
Root discriminant \(332.69\)
Ramified primes $2,191$
Class number not computed
Class group not computed
Galois group $C_{38}$ (as 38T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^38 - 191*x^36 + 15662*x^34 - 730575*x^32 + 21705622*x^30 - 436357836*x^28 + 6157331558*x^26 - 62411853712*x^24 + 460944322269*x^22 - 2499589479956*x^20 + 9974553348782*x^18 - 29202012585177*x^16 + 62167403988174*x^14 - 94680464008195*x^12 + 100459885220950*x^10 - 71207905317537*x^8 + 31469275400290*x^6 - 7629511512714*x^4 + 744125807285*x^2 - 1101076991)
 
gp: K = bnfinit(y^38 - 191*y^36 + 15662*y^34 - 730575*y^32 + 21705622*y^30 - 436357836*y^28 + 6157331558*y^26 - 62411853712*y^24 + 460944322269*y^22 - 2499589479956*y^20 + 9974553348782*y^18 - 29202012585177*y^16 + 62167403988174*y^14 - 94680464008195*y^12 + 100459885220950*y^10 - 71207905317537*y^8 + 31469275400290*y^6 - 7629511512714*y^4 + 744125807285*y^2 - 1101076991, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^38 - 191*x^36 + 15662*x^34 - 730575*x^32 + 21705622*x^30 - 436357836*x^28 + 6157331558*x^26 - 62411853712*x^24 + 460944322269*x^22 - 2499589479956*x^20 + 9974553348782*x^18 - 29202012585177*x^16 + 62167403988174*x^14 - 94680464008195*x^12 + 100459885220950*x^10 - 71207905317537*x^8 + 31469275400290*x^6 - 7629511512714*x^4 + 744125807285*x^2 - 1101076991);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^38 - 191*x^36 + 15662*x^34 - 730575*x^32 + 21705622*x^30 - 436357836*x^28 + 6157331558*x^26 - 62411853712*x^24 + 460944322269*x^22 - 2499589479956*x^20 + 9974553348782*x^18 - 29202012585177*x^16 + 62167403988174*x^14 - 94680464008195*x^12 + 100459885220950*x^10 - 71207905317537*x^8 + 31469275400290*x^6 - 7629511512714*x^4 + 744125807285*x^2 - 1101076991)
 

\( x^{38} - 191 x^{36} + 15662 x^{34} - 730575 x^{32} + 21705622 x^{30} - 436357836 x^{28} + \cdots - 1101076991 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $38$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[38, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(687\!\cdots\!664\) \(\medspace = 2^{38}\cdot 191^{37}\) 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:  \(332.69\)
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:  $2\cdot 191^{37/38}\approx 332.6872590953789$
Ramified primes:   \(2\), \(191\) 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:  \(\Q(\sqrt{191}) \)
$\card{ \Gal(K/\Q) }$:  $38$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(764=2^{2}\cdot 191\)
Dirichlet character group:    $\lbrace$$\chi_{764}(1,·)$, $\chi_{764}(643,·)$, $\chi_{764}(5,·)$, $\chi_{764}(567,·)$, $\chi_{764}(139,·)$, $\chi_{764}(275,·)$, $\chi_{764}(25,·)$, $\chi_{764}(153,·)$, $\chi_{764}(155,·)$, $\chi_{764}(31,·)$, $\chi_{764}(197,·)$, $\chi_{764}(419,·)$, $\chi_{764}(423,·)$, $\chi_{764}(605,·)$, $\chi_{764}(177,·)$, $\chi_{764}(709,·)$, $\chi_{764}(695,·)$, $\chi_{764}(543,·)$, $\chi_{764}(159,·)$, $\chi_{764}(11,·)$, $\chi_{764}(69,·)$, $\chi_{764}(55,·)$, $\chi_{764}(611,·)$, $\chi_{764}(341,·)$, $\chi_{764}(625,·)$, $\chi_{764}(345,·)$, $\chi_{764}(733,·)$, $\chi_{764}(609,·)$, $\chi_{764}(587,·)$, $\chi_{764}(739,·)$, $\chi_{764}(489,·)$, $\chi_{764}(753,·)$, $\chi_{764}(221,·)$, $\chi_{764}(759,·)$, $\chi_{764}(121,·)$, $\chi_{764}(763,·)$, $\chi_{764}(125,·)$, $\chi_{764}(639,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{2}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{3}$, $\frac{1}{7}a^{10}-\frac{1}{7}a^{4}$, $\frac{1}{7}a^{11}-\frac{1}{7}a^{5}$, $\frac{1}{7}a^{12}-\frac{1}{7}a^{6}$, $\frac{1}{7}a^{13}-\frac{1}{7}a$, $\frac{1}{49}a^{14}-\frac{2}{49}a^{8}+\frac{1}{49}a^{2}$, $\frac{1}{49}a^{15}-\frac{2}{49}a^{9}+\frac{1}{49}a^{3}$, $\frac{1}{49}a^{16}-\frac{2}{49}a^{10}+\frac{1}{49}a^{4}$, $\frac{1}{49}a^{17}-\frac{2}{49}a^{11}+\frac{1}{49}a^{5}$, $\frac{1}{49}a^{18}-\frac{2}{49}a^{12}+\frac{1}{49}a^{6}$, $\frac{1}{49}a^{19}-\frac{2}{49}a^{13}+\frac{1}{49}a^{7}$, $\frac{1}{49}a^{20}-\frac{3}{49}a^{8}+\frac{2}{49}a^{2}$, $\frac{1}{343}a^{21}-\frac{3}{343}a^{15}+\frac{3}{343}a^{9}-\frac{1}{343}a^{3}$, $\frac{1}{343}a^{22}-\frac{3}{343}a^{16}+\frac{3}{343}a^{10}-\frac{1}{343}a^{4}$, $\frac{1}{343}a^{23}-\frac{3}{343}a^{17}+\frac{3}{343}a^{11}-\frac{1}{343}a^{5}$, $\frac{1}{2401}a^{24}+\frac{3}{2401}a^{22}+\frac{1}{343}a^{20}+\frac{4}{2401}a^{18}-\frac{2}{2401}a^{16}-\frac{2}{343}a^{14}+\frac{87}{2401}a^{12}+\frac{93}{2401}a^{10}+\frac{15}{343}a^{8}-\frac{778}{2401}a^{6}+\frac{935}{2401}a^{4}-\frac{9}{49}a^{2}$, $\frac{1}{2401}a^{25}+\frac{3}{2401}a^{23}+\frac{4}{2401}a^{19}-\frac{2}{2401}a^{17}+\frac{1}{343}a^{15}+\frac{87}{2401}a^{13}+\frac{93}{2401}a^{11}+\frac{12}{343}a^{9}-\frac{92}{2401}a^{7}+\frac{935}{2401}a^{5}-\frac{62}{343}a^{3}-\frac{2}{7}a$, $\frac{1}{2401}a^{26}-\frac{2}{2401}a^{22}-\frac{17}{2401}a^{20}-\frac{2}{343}a^{18}-\frac{8}{2401}a^{16}-\frac{18}{2401}a^{14}-\frac{24}{343}a^{12}+\frac{169}{2401}a^{10}-\frac{113}{2401}a^{8}+\frac{124}{343}a^{6}-\frac{1188}{2401}a^{4}+\frac{10}{49}a^{2}$, $\frac{1}{2401}a^{27}-\frac{2}{2401}a^{23}-\frac{3}{2401}a^{21}-\frac{2}{343}a^{19}-\frac{8}{2401}a^{17}-\frac{11}{2401}a^{15}-\frac{24}{343}a^{13}+\frac{169}{2401}a^{11}-\frac{169}{2401}a^{9}-\frac{23}{343}a^{7}-\frac{1188}{2401}a^{5}+\frac{75}{343}a^{3}+\frac{3}{7}a$, $\frac{1}{1831963}a^{28}+\frac{22}{261709}a^{26}+\frac{36}{261709}a^{24}+\frac{2096}{1831963}a^{22}+\frac{1418}{261709}a^{20}-\frac{2250}{261709}a^{18}-\frac{14967}{1831963}a^{16}-\frac{1915}{261709}a^{14}-\frac{13738}{261709}a^{12}-\frac{22320}{1831963}a^{10}+\frac{9582}{261709}a^{8}+\frac{27957}{261709}a^{6}+\frac{20784}{1831963}a^{4}-\frac{13747}{37387}a^{2}-\frac{202}{763}$, $\frac{1}{1831963}a^{29}+\frac{22}{261709}a^{27}+\frac{36}{261709}a^{25}+\frac{2096}{1831963}a^{23}-\frac{108}{261709}a^{21}-\frac{2250}{261709}a^{19}-\frac{14967}{1831963}a^{17}+\frac{2663}{261709}a^{15}-\frac{13738}{261709}a^{13}-\frac{22320}{1831963}a^{11}+\frac{5004}{261709}a^{9}-\frac{9430}{261709}a^{7}+\frac{20784}{1831963}a^{5}-\frac{13529}{37387}a^{3}-\frac{93}{763}a$, $\frac{1}{1831963}a^{30}+\frac{27}{261709}a^{26}-\frac{88}{1831963}a^{24}+\frac{15}{37387}a^{22}-\frac{1096}{261709}a^{20}-\frac{17333}{1831963}a^{18}+\frac{2}{763}a^{16}-\frac{2446}{261709}a^{14}-\frac{95071}{1831963}a^{12}+\frac{1446}{37387}a^{10}+\frac{17725}{261709}a^{8}+\frac{242145}{1831963}a^{6}+\frac{4566}{37387}a^{4}-\frac{2327}{5341}a^{2}-\frac{25}{109}$, $\frac{1}{12823741}a^{31}+\frac{3}{12823741}a^{29}+\frac{93}{1831963}a^{27}-\frac{95}{12823741}a^{25}-\frac{16630}{12823741}a^{23}+\frac{2395}{1831963}a^{21}+\frac{44526}{12823741}a^{19}-\frac{49255}{12823741}a^{17}+\frac{14699}{1831963}a^{15}+\frac{634273}{12823741}a^{13}+\frac{803518}{12823741}a^{11}-\frac{45089}{1831963}a^{9}+\frac{226472}{12823741}a^{7}+\frac{6061996}{12823741}a^{5}+\frac{32504}{261709}a^{3}-\frac{781}{5341}a$, $\frac{1}{12823741}a^{32}+\frac{3}{12823741}a^{30}+\frac{1130}{12823741}a^{26}+\frac{912}{12823741}a^{24}+\frac{506}{1831963}a^{22}-\frac{66851}{12823741}a^{20}+\frac{88141}{12823741}a^{18}-\frac{16365}{1831963}a^{16}+\frac{19519}{12823741}a^{14}+\frac{394466}{12823741}a^{12}+\frac{72813}{1831963}a^{10}-\frac{507744}{12823741}a^{8}-\frac{759637}{12823741}a^{6}+\frac{435594}{1831963}a^{4}-\frac{8073}{37387}a^{2}-\frac{289}{763}$, $\frac{1}{12823741}a^{33}-\frac{2}{12823741}a^{29}+\frac{255}{12823741}a^{27}-\frac{340}{1831963}a^{25}+\frac{14694}{12823741}a^{23}-\frac{10277}{12823741}a^{21}+\frac{12094}{1831963}a^{19}+\frac{51284}{12823741}a^{17}-\frac{9125}{12823741}a^{15}+\frac{70618}{1831963}a^{13}-\frac{834014}{12823741}a^{11}-\frac{474676}{12823741}a^{9}+\frac{97703}{1831963}a^{7}+\frac{3867729}{12823741}a^{5}+\frac{68682}{261709}a^{3}-\frac{1094}{5341}a$, $\frac{1}{686313794579}a^{34}-\frac{19890}{686313794579}a^{32}-\frac{45819}{686313794579}a^{30}+\frac{5876}{686313794579}a^{28}-\frac{113957209}{686313794579}a^{26}+\frac{135161525}{686313794579}a^{24}+\frac{64903026}{686313794579}a^{22}+\frac{1815319684}{686313794579}a^{20}+\frac{904338957}{686313794579}a^{18}+\frac{6515432227}{686313794579}a^{16}+\frac{3123289335}{686313794579}a^{14}+\frac{25923280105}{686313794579}a^{12}-\frac{10106665799}{686313794579}a^{10}+\frac{14261364695}{686313794579}a^{8}-\frac{138914368254}{686313794579}a^{6}-\frac{22900353157}{98044827797}a^{4}-\frac{567990517}{2000914853}a^{2}-\frac{13811668}{40834997}$, $\frac{1}{686313794579}a^{35}-\frac{19890}{686313794579}a^{33}+\frac{1100}{98044827797}a^{31}+\frac{166433}{686313794579}a^{29}-\frac{79116340}{686313794579}a^{27}+\frac{18582460}{98044827797}a^{25}-\frac{117873992}{98044827797}a^{23}+\frac{711650866}{686313794579}a^{21}+\frac{469617993}{98044827797}a^{19}+\frac{3879353882}{686313794579}a^{17}+\frac{626360390}{686313794579}a^{15}-\frac{5453698715}{98044827797}a^{13}+\frac{32896814043}{686313794579}a^{11}+\frac{19379600741}{686313794579}a^{9}-\frac{4106997927}{98044827797}a^{7}+\frac{164129491825}{686313794579}a^{5}-\frac{2481362025}{14006403971}a^{3}-\frac{138480015}{285844979}a$, $\frac{1}{20\!\cdots\!07}a^{36}+\frac{11\!\cdots\!41}{20\!\cdots\!07}a^{34}+\frac{24\!\cdots\!46}{20\!\cdots\!07}a^{32}-\frac{10\!\cdots\!46}{42\!\cdots\!77}a^{30}-\frac{44\!\cdots\!42}{20\!\cdots\!07}a^{28}+\frac{39\!\cdots\!22}{20\!\cdots\!07}a^{26}-\frac{27\!\cdots\!89}{29\!\cdots\!01}a^{24}+\frac{21\!\cdots\!04}{20\!\cdots\!07}a^{22}-\frac{18\!\cdots\!54}{20\!\cdots\!07}a^{20}-\frac{12\!\cdots\!25}{20\!\cdots\!07}a^{18}+\frac{15\!\cdots\!33}{20\!\cdots\!07}a^{16}-\frac{12\!\cdots\!10}{20\!\cdots\!07}a^{14}+\frac{12\!\cdots\!99}{20\!\cdots\!07}a^{12}-\frac{12\!\cdots\!95}{20\!\cdots\!07}a^{10}-\frac{11\!\cdots\!27}{20\!\cdots\!07}a^{8}+\frac{28\!\cdots\!75}{20\!\cdots\!07}a^{6}-\frac{12\!\cdots\!37}{42\!\cdots\!43}a^{4}+\frac{11\!\cdots\!80}{85\!\cdots\!07}a^{2}-\frac{69\!\cdots\!21}{17\!\cdots\!43}$, $\frac{1}{14\!\cdots\!49}a^{37}-\frac{48\!\cdots\!25}{14\!\cdots\!49}a^{35}-\frac{38\!\cdots\!41}{14\!\cdots\!49}a^{33}+\frac{38\!\cdots\!76}{14\!\cdots\!49}a^{31}-\frac{16\!\cdots\!47}{14\!\cdots\!49}a^{29}-\frac{28\!\cdots\!07}{14\!\cdots\!49}a^{27}-\frac{96\!\cdots\!50}{20\!\cdots\!07}a^{25}+\frac{91\!\cdots\!02}{14\!\cdots\!49}a^{23}-\frac{16\!\cdots\!61}{14\!\cdots\!49}a^{21}+\frac{10\!\cdots\!25}{14\!\cdots\!49}a^{19}+\frac{43\!\cdots\!87}{14\!\cdots\!49}a^{17}+\frac{87\!\cdots\!94}{13\!\cdots\!61}a^{15}-\frac{91\!\cdots\!21}{14\!\cdots\!49}a^{13}+\frac{45\!\cdots\!89}{14\!\cdots\!49}a^{11}-\frac{21\!\cdots\!66}{14\!\cdots\!49}a^{9}+\frac{96\!\cdots\!79}{14\!\cdots\!49}a^{7}+\frac{13\!\cdots\!31}{29\!\cdots\!01}a^{5}+\frac{97\!\cdots\!19}{60\!\cdots\!49}a^{3}-\frac{40\!\cdots\!90}{12\!\cdots\!01}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $7$

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:  $37$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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^38 - 191*x^36 + 15662*x^34 - 730575*x^32 + 21705622*x^30 - 436357836*x^28 + 6157331558*x^26 - 62411853712*x^24 + 460944322269*x^22 - 2499589479956*x^20 + 9974553348782*x^18 - 29202012585177*x^16 + 62167403988174*x^14 - 94680464008195*x^12 + 100459885220950*x^10 - 71207905317537*x^8 + 31469275400290*x^6 - 7629511512714*x^4 + 744125807285*x^2 - 1101076991)
 
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^38 - 191*x^36 + 15662*x^34 - 730575*x^32 + 21705622*x^30 - 436357836*x^28 + 6157331558*x^26 - 62411853712*x^24 + 460944322269*x^22 - 2499589479956*x^20 + 9974553348782*x^18 - 29202012585177*x^16 + 62167403988174*x^14 - 94680464008195*x^12 + 100459885220950*x^10 - 71207905317537*x^8 + 31469275400290*x^6 - 7629511512714*x^4 + 744125807285*x^2 - 1101076991, 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^38 - 191*x^36 + 15662*x^34 - 730575*x^32 + 21705622*x^30 - 436357836*x^28 + 6157331558*x^26 - 62411853712*x^24 + 460944322269*x^22 - 2499589479956*x^20 + 9974553348782*x^18 - 29202012585177*x^16 + 62167403988174*x^14 - 94680464008195*x^12 + 100459885220950*x^10 - 71207905317537*x^8 + 31469275400290*x^6 - 7629511512714*x^4 + 744125807285*x^2 - 1101076991);
 
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^38 - 191*x^36 + 15662*x^34 - 730575*x^32 + 21705622*x^30 - 436357836*x^28 + 6157331558*x^26 - 62411853712*x^24 + 460944322269*x^22 - 2499589479956*x^20 + 9974553348782*x^18 - 29202012585177*x^16 + 62167403988174*x^14 - 94680464008195*x^12 + 100459885220950*x^10 - 71207905317537*x^8 + 31469275400290*x^6 - 7629511512714*x^4 + 744125807285*x^2 - 1101076991);
 
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

$C_{38}$ (as 38T1):

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 38
The 38 conjugacy class representatives for $C_{38}$
Character table for $C_{38}$

Intermediate fields

\(\Q(\sqrt{191}) \), 19.19.114445997944945591651333831028437092270721.1

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 $38$ $19^{2}$ ${\href{/padicField/7.1.0.1}{1} }^{38}$ $19^{2}$ $19^{2}$ $19^{2}$ $19^{2}$ $38$ $38$ $19^{2}$ $38$ $38$ $38$ $19^{2}$ $38$ $38$

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
\(2\) Copy content Toggle raw display Deg $38$$2$$19$$38$
\(191\) Copy content Toggle raw display Deg $38$$38$$1$$37$