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Properties

Label 35.35.705...625.1
Degree $35$
Signature $[35, 0]$
Discriminant $7.050\times 10^{89}$
Root discriminant \(369.06\)
Ramified primes $5,7$
Class number not computed
Class group not computed
Galois group $C_{35}$ (as 35T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^35 - 175*x^33 - 70*x^32 + 13125*x^31 + 9702*x^30 - 556640*x^29 - 573225*x^28 + 14844900*x^27 + 19050185*x^26 - 261854635*x^25 - 394847250*x^24 + 3128435905*x^23 + 5347852195*x^22 - 25515383355*x^21 - 48297678121*x^20 + 141795944465*x^19 + 292585456625*x^18 - 533394487115*x^17 - 1187150912745*x^16 + 1349113537702*x^15 + 3207931617015*x^14 - 2286376450785*x^13 - 5710072444965*x^12 + 2606036871270*x^11 + 6560227221870*x^10 - 2026242763720*x^9 - 4691478893710*x^8 + 1084067118185*x^7 + 1962115542205*x^6 - 373976828869*x^5 - 432767079290*x^4 + 65717425060*x^3 + 43746645075*x^2 - 3865111915*x - 1645272349)
 
gp: K = bnfinit(y^35 - 175*y^33 - 70*y^32 + 13125*y^31 + 9702*y^30 - 556640*y^29 - 573225*y^28 + 14844900*y^27 + 19050185*y^26 - 261854635*y^25 - 394847250*y^24 + 3128435905*y^23 + 5347852195*y^22 - 25515383355*y^21 - 48297678121*y^20 + 141795944465*y^19 + 292585456625*y^18 - 533394487115*y^17 - 1187150912745*y^16 + 1349113537702*y^15 + 3207931617015*y^14 - 2286376450785*y^13 - 5710072444965*y^12 + 2606036871270*y^11 + 6560227221870*y^10 - 2026242763720*y^9 - 4691478893710*y^8 + 1084067118185*y^7 + 1962115542205*y^6 - 373976828869*y^5 - 432767079290*y^4 + 65717425060*y^3 + 43746645075*y^2 - 3865111915*y - 1645272349, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^35 - 175*x^33 - 70*x^32 + 13125*x^31 + 9702*x^30 - 556640*x^29 - 573225*x^28 + 14844900*x^27 + 19050185*x^26 - 261854635*x^25 - 394847250*x^24 + 3128435905*x^23 + 5347852195*x^22 - 25515383355*x^21 - 48297678121*x^20 + 141795944465*x^19 + 292585456625*x^18 - 533394487115*x^17 - 1187150912745*x^16 + 1349113537702*x^15 + 3207931617015*x^14 - 2286376450785*x^13 - 5710072444965*x^12 + 2606036871270*x^11 + 6560227221870*x^10 - 2026242763720*x^9 - 4691478893710*x^8 + 1084067118185*x^7 + 1962115542205*x^6 - 373976828869*x^5 - 432767079290*x^4 + 65717425060*x^3 + 43746645075*x^2 - 3865111915*x - 1645272349);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^35 - 175*x^33 - 70*x^32 + 13125*x^31 + 9702*x^30 - 556640*x^29 - 573225*x^28 + 14844900*x^27 + 19050185*x^26 - 261854635*x^25 - 394847250*x^24 + 3128435905*x^23 + 5347852195*x^22 - 25515383355*x^21 - 48297678121*x^20 + 141795944465*x^19 + 292585456625*x^18 - 533394487115*x^17 - 1187150912745*x^16 + 1349113537702*x^15 + 3207931617015*x^14 - 2286376450785*x^13 - 5710072444965*x^12 + 2606036871270*x^11 + 6560227221870*x^10 - 2026242763720*x^9 - 4691478893710*x^8 + 1084067118185*x^7 + 1962115542205*x^6 - 373976828869*x^5 - 432767079290*x^4 + 65717425060*x^3 + 43746645075*x^2 - 3865111915*x - 1645272349)
 

\( x^{35} - 175 x^{33} - 70 x^{32} + 13125 x^{31} + 9702 x^{30} - 556640 x^{29} - 573225 x^{28} + \cdots - 1645272349 \) Copy content Toggle raw display

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

Invariants

Degree:  $35$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[35, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(705\!\cdots\!625\) \(\medspace = 5^{56}\cdot 7^{60}\) 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:  \(369.06\)
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:  $5^{8/5}7^{12/7}\approx 369.0553488603634$
Ramified primes:   \(5\), \(7\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $35$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1225=5^{2}\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{1225}(1,·)$, $\chi_{1225}(386,·)$, $\chi_{1225}(771,·)$, $\chi_{1225}(1156,·)$, $\chi_{1225}(141,·)$, $\chi_{1225}(526,·)$, $\chi_{1225}(911,·)$, $\chi_{1225}(281,·)$, $\chi_{1225}(666,·)$, $\chi_{1225}(1051,·)$, $\chi_{1225}(36,·)$, $\chi_{1225}(421,·)$, $\chi_{1225}(806,·)$, $\chi_{1225}(1191,·)$, $\chi_{1225}(176,·)$, $\chi_{1225}(561,·)$, $\chi_{1225}(946,·)$, $\chi_{1225}(316,·)$, $\chi_{1225}(701,·)$, $\chi_{1225}(1086,·)$, $\chi_{1225}(71,·)$, $\chi_{1225}(456,·)$, $\chi_{1225}(841,·)$, $\chi_{1225}(211,·)$, $\chi_{1225}(596,·)$, $\chi_{1225}(981,·)$, $\chi_{1225}(351,·)$, $\chi_{1225}(736,·)$, $\chi_{1225}(1121,·)$, $\chi_{1225}(106,·)$, $\chi_{1225}(491,·)$, $\chi_{1225}(876,·)$, $\chi_{1225}(246,·)$, $\chi_{1225}(631,·)$, $\chi_{1225}(1016,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{49}a^{28}+\frac{3}{7}a^{27}-\frac{1}{7}a^{25}-\frac{3}{7}a^{24}+\frac{3}{7}a^{22}+\frac{1}{7}a^{21}+\frac{2}{7}a^{19}-\frac{3}{7}a^{18}-\frac{2}{7}a^{16}-\frac{3}{7}a^{15}-\frac{3}{7}a^{13}-\frac{3}{7}a^{12}+\frac{1}{7}a^{10}-\frac{3}{7}a^{9}-\frac{1}{49}a^{7}+\frac{1}{7}a^{6}-\frac{1}{7}$, $\frac{1}{49}a^{29}-\frac{1}{7}a^{26}-\frac{3}{7}a^{25}+\frac{3}{7}a^{23}+\frac{1}{7}a^{22}+\frac{2}{7}a^{20}-\frac{3}{7}a^{19}-\frac{2}{7}a^{17}-\frac{3}{7}a^{16}-\frac{3}{7}a^{14}-\frac{3}{7}a^{13}+\frac{1}{7}a^{11}-\frac{3}{7}a^{10}-\frac{1}{49}a^{8}-\frac{3}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{202027}a^{30}+\frac{17}{202027}a^{29}-\frac{934}{202027}a^{28}-\frac{185}{28861}a^{27}-\frac{4535}{28861}a^{26}+\frac{3018}{28861}a^{25}+\frac{1244}{28861}a^{24}+\frac{13387}{28861}a^{23}+\frac{5391}{28861}a^{22}-\frac{4684}{28861}a^{21}+\frac{4777}{28861}a^{20}-\frac{4803}{28861}a^{19}+\frac{537}{4123}a^{18}-\frac{5511}{28861}a^{17}-\frac{13233}{28861}a^{16}+\frac{459}{1519}a^{15}-\frac{414}{1519}a^{14}-\frac{191}{4123}a^{13}-\frac{12793}{28861}a^{12}+\frac{1294}{4123}a^{11}+\frac{223}{1519}a^{10}-\frac{1851}{10633}a^{9}+\frac{2546}{10633}a^{8}-\frac{52441}{202027}a^{7}-\frac{6394}{28861}a^{6}-\frac{905}{4123}a^{5}-\frac{142}{4123}a^{4}+\frac{20}{217}a^{3}+\frac{3030}{28861}a^{2}-\frac{12757}{28861}a+\frac{9915}{28861}$, $\frac{1}{202027}a^{31}-\frac{1223}{202027}a^{29}-\frac{1909}{202027}a^{28}+\frac{6856}{28861}a^{27}-\frac{6470}{28861}a^{26}-\frac{4709}{28861}a^{25}+\frac{12854}{28861}a^{24}+\frac{8700}{28861}a^{23}-\frac{1502}{28861}a^{22}+\frac{10191}{28861}a^{21}+\frac{571}{28861}a^{20}-\frac{5296}{28861}a^{19}+\frac{8923}{28861}a^{18}-\frac{6129}{28861}a^{17}+\frac{6917}{28861}a^{16}+\frac{463}{1519}a^{15}-\frac{11920}{28861}a^{14}+\frac{1690}{28861}a^{13}-\frac{12595}{28861}a^{12}-\frac{5444}{28861}a^{11}-\frac{2565}{10633}a^{10}-\frac{132}{1519}a^{9}-\frac{66691}{202027}a^{8}+\frac{55123}{202027}a^{7}-\frac{712}{28861}a^{6}-\frac{1249}{4123}a^{5}-\frac{1329}{4123}a^{4}-\frac{13329}{28861}a^{3}-\frac{935}{4123}a^{2}-\frac{216}{1519}a-\frac{7758}{28861}$, $\frac{1}{202027}a^{32}-\frac{1733}{202027}a^{29}-\frac{1695}{202027}a^{28}-\frac{14206}{28861}a^{27}+\frac{1559}{4123}a^{26}-\frac{1565}{4123}a^{25}+\frac{12848}{28861}a^{24}+\frac{131}{1519}a^{23}-\frac{9908}{28861}a^{22}+\frac{11255}{28861}a^{21}-\frac{5316}{28861}a^{20}-\frac{214}{589}a^{19}-\frac{14263}{28861}a^{18}+\frac{3946}{28861}a^{17}-\frac{633}{28861}a^{16}-\frac{398}{931}a^{15}-\frac{3592}{28861}a^{14}+\frac{13823}{28861}a^{13}+\frac{3748}{28861}a^{12}-\frac{23843}{202027}a^{11}+\frac{696}{1519}a^{10}+\frac{5708}{28861}a^{9}+\frac{43229}{202027}a^{8}+\frac{50660}{202027}a^{7}-\frac{11397}{28861}a^{6}+\frac{943}{4123}a^{5}+\frac{12032}{28861}a^{4}+\frac{2029}{4123}a^{3}+\frac{34}{133}a^{2}-\frac{4014}{28861}a+\frac{8548}{28861}$, $\frac{1}{122630389}a^{33}-\frac{4}{122630389}a^{32}-\frac{281}{122630389}a^{31}-\frac{9}{3955819}a^{30}-\frac{103889}{17518627}a^{29}+\frac{1200830}{122630389}a^{28}-\frac{8308166}{17518627}a^{27}-\frac{9245}{18817}a^{26}+\frac{750378}{17518627}a^{25}+\frac{449578}{922033}a^{24}+\frac{2268655}{17518627}a^{23}+\frac{2444434}{17518627}a^{22}-\frac{2587630}{17518627}a^{21}-\frac{6372777}{17518627}a^{20}-\frac{1340582}{17518627}a^{19}-\frac{93644}{17518627}a^{18}-\frac{3210789}{17518627}a^{17}+\frac{1024582}{17518627}a^{16}+\frac{243493}{17518627}a^{15}+\frac{1967683}{17518627}a^{14}+\frac{4790171}{17518627}a^{13}+\frac{58513209}{122630389}a^{12}+\frac{16188141}{122630389}a^{11}-\frac{17309172}{122630389}a^{10}-\frac{55381439}{122630389}a^{9}-\frac{231898}{2502661}a^{8}+\frac{54440133}{122630389}a^{7}+\frac{5971351}{17518627}a^{6}+\frac{821764}{17518627}a^{5}+\frac{2260521}{17518627}a^{4}+\frac{1358540}{17518627}a^{3}+\frac{92707}{17518627}a^{2}+\frac{2000}{11533}a+\frac{8068836}{17518627}$, $\frac{1}{10\!\cdots\!11}a^{34}+\frac{17\!\cdots\!07}{10\!\cdots\!11}a^{33}-\frac{74\!\cdots\!82}{10\!\cdots\!11}a^{32}+\frac{47\!\cdots\!17}{10\!\cdots\!11}a^{31}-\frac{11\!\cdots\!33}{10\!\cdots\!11}a^{30}-\frac{24\!\cdots\!41}{10\!\cdots\!11}a^{29}+\frac{49\!\cdots\!55}{56\!\cdots\!69}a^{28}-\frac{71\!\cdots\!95}{15\!\cdots\!73}a^{27}-\frac{10\!\cdots\!08}{22\!\cdots\!39}a^{26}-\frac{30\!\cdots\!83}{15\!\cdots\!73}a^{25}+\frac{13\!\cdots\!12}{15\!\cdots\!73}a^{24}-\frac{35\!\cdots\!10}{15\!\cdots\!73}a^{23}-\frac{77\!\cdots\!66}{15\!\cdots\!73}a^{22}+\frac{67\!\cdots\!94}{15\!\cdots\!73}a^{21}-\frac{58\!\cdots\!33}{15\!\cdots\!73}a^{20}+\frac{16\!\cdots\!85}{15\!\cdots\!73}a^{19}+\frac{17\!\cdots\!74}{15\!\cdots\!73}a^{18}-\frac{40\!\cdots\!53}{15\!\cdots\!73}a^{17}+\frac{70\!\cdots\!74}{15\!\cdots\!73}a^{16}-\frac{75\!\cdots\!19}{15\!\cdots\!73}a^{15}+\frac{71\!\cdots\!13}{15\!\cdots\!73}a^{14}+\frac{36\!\cdots\!34}{10\!\cdots\!11}a^{13}+\frac{95\!\cdots\!92}{10\!\cdots\!11}a^{12}-\frac{50\!\cdots\!20}{10\!\cdots\!11}a^{11}+\frac{20\!\cdots\!70}{10\!\cdots\!11}a^{10}-\frac{11\!\cdots\!57}{56\!\cdots\!69}a^{9}-\frac{17\!\cdots\!86}{10\!\cdots\!11}a^{8}+\frac{46\!\cdots\!42}{10\!\cdots\!11}a^{7}+\frac{24\!\cdots\!37}{15\!\cdots\!73}a^{6}-\frac{70\!\cdots\!03}{15\!\cdots\!73}a^{5}+\frac{36\!\cdots\!76}{15\!\cdots\!73}a^{4}+\frac{11\!\cdots\!17}{81\!\cdots\!67}a^{3}-\frac{25\!\cdots\!52}{15\!\cdots\!73}a^{2}+\frac{43\!\cdots\!16}{15\!\cdots\!73}a+\frac{14\!\cdots\!85}{49\!\cdots\!83}$ Copy content Toggle raw display

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:  $34$
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^35 - 175*x^33 - 70*x^32 + 13125*x^31 + 9702*x^30 - 556640*x^29 - 573225*x^28 + 14844900*x^27 + 19050185*x^26 - 261854635*x^25 - 394847250*x^24 + 3128435905*x^23 + 5347852195*x^22 - 25515383355*x^21 - 48297678121*x^20 + 141795944465*x^19 + 292585456625*x^18 - 533394487115*x^17 - 1187150912745*x^16 + 1349113537702*x^15 + 3207931617015*x^14 - 2286376450785*x^13 - 5710072444965*x^12 + 2606036871270*x^11 + 6560227221870*x^10 - 2026242763720*x^9 - 4691478893710*x^8 + 1084067118185*x^7 + 1962115542205*x^6 - 373976828869*x^5 - 432767079290*x^4 + 65717425060*x^3 + 43746645075*x^2 - 3865111915*x - 1645272349)
 
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^35 - 175*x^33 - 70*x^32 + 13125*x^31 + 9702*x^30 - 556640*x^29 - 573225*x^28 + 14844900*x^27 + 19050185*x^26 - 261854635*x^25 - 394847250*x^24 + 3128435905*x^23 + 5347852195*x^22 - 25515383355*x^21 - 48297678121*x^20 + 141795944465*x^19 + 292585456625*x^18 - 533394487115*x^17 - 1187150912745*x^16 + 1349113537702*x^15 + 3207931617015*x^14 - 2286376450785*x^13 - 5710072444965*x^12 + 2606036871270*x^11 + 6560227221870*x^10 - 2026242763720*x^9 - 4691478893710*x^8 + 1084067118185*x^7 + 1962115542205*x^6 - 373976828869*x^5 - 432767079290*x^4 + 65717425060*x^3 + 43746645075*x^2 - 3865111915*x - 1645272349, 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^35 - 175*x^33 - 70*x^32 + 13125*x^31 + 9702*x^30 - 556640*x^29 - 573225*x^28 + 14844900*x^27 + 19050185*x^26 - 261854635*x^25 - 394847250*x^24 + 3128435905*x^23 + 5347852195*x^22 - 25515383355*x^21 - 48297678121*x^20 + 141795944465*x^19 + 292585456625*x^18 - 533394487115*x^17 - 1187150912745*x^16 + 1349113537702*x^15 + 3207931617015*x^14 - 2286376450785*x^13 - 5710072444965*x^12 + 2606036871270*x^11 + 6560227221870*x^10 - 2026242763720*x^9 - 4691478893710*x^8 + 1084067118185*x^7 + 1962115542205*x^6 - 373976828869*x^5 - 432767079290*x^4 + 65717425060*x^3 + 43746645075*x^2 - 3865111915*x - 1645272349);
 
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^35 - 175*x^33 - 70*x^32 + 13125*x^31 + 9702*x^30 - 556640*x^29 - 573225*x^28 + 14844900*x^27 + 19050185*x^26 - 261854635*x^25 - 394847250*x^24 + 3128435905*x^23 + 5347852195*x^22 - 25515383355*x^21 - 48297678121*x^20 + 141795944465*x^19 + 292585456625*x^18 - 533394487115*x^17 - 1187150912745*x^16 + 1349113537702*x^15 + 3207931617015*x^14 - 2286376450785*x^13 - 5710072444965*x^12 + 2606036871270*x^11 + 6560227221870*x^10 - 2026242763720*x^9 - 4691478893710*x^8 + 1084067118185*x^7 + 1962115542205*x^6 - 373976828869*x^5 - 432767079290*x^4 + 65717425060*x^3 + 43746645075*x^2 - 3865111915*x - 1645272349);
 
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

$C_{35}$ (as 35T1):

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 35
The 35 conjugacy class representatives for $C_{35}$
Character table for $C_{35}$ is not computed

Intermediate fields

5.5.390625.1, 7.7.13841287201.1

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 $35$ $35$ R R $35$ $35$ $35$ ${\href{/padicField/19.5.0.1}{5} }^{7}$ $35$ $35$ ${\href{/padicField/31.5.0.1}{5} }^{7}$ $35$ $35$ ${\href{/padicField/43.7.0.1}{7} }^{5}$ $35$ $35$ $35$

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
\(5\) Copy content Toggle raw display Deg $35$$5$$7$$56$
\(7\) Copy content Toggle raw display 7.7.12.1$x^{7} + 42 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} + 42 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} + 42 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} + 42 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} + 42 x^{6} + 7$$7$$1$$12$$C_7$$[2]$