| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 47.1-a1 |
47.1-a |
$2$ |
$5$ |
6.6.1081856.1 |
$6$ |
$[6, 0]$ |
47.1 |
\( 47 \) |
\( -47 \) |
$128.10461$ |
$(2a^5-a^4-10a^3+8a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$625$ |
\( 1 \) |
$4.667511436$ |
$0.191559365$ |
3.22356 |
\( \frac{7311665291966906351947241528318565373890469}{47} a^{5} - \frac{4346954806112764409148136193536185564093647}{47} a^{4} - 878417578200993987253098866163900427396856 a^{3} + \frac{9921932903976042368279319434408749783081233}{47} a^{2} + \frac{45282837527200374048911690129686890606008500}{47} a - \frac{12298367875041277280162183332857563285605726}{47} \) |
\( \bigl[a^{5} - a^{4} - 5 a^{3} + 2 a^{2} + 6 a\) , \( a^{5} - 2 a^{4} - 3 a^{3} + 5 a^{2} - 1\) , \( a^{4} - 5 a^{2} - a + 4\) , \( 915 a^{5} - 493 a^{4} - 5119 a^{3} + 938 a^{2} + 5380 a - 1450\) , \( 20553 a^{5} - 11707 a^{4} - 115726 a^{3} + 25140 a^{2} + 124898 a - 33781\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-5a^{3}+2a^{2}+6a\right){x}{y}+\left(a^{4}-5a^{2}-a+4\right){y}={x}^{3}+\left(a^{5}-2a^{4}-3a^{3}+5a^{2}-1\right){x}^{2}+\left(915a^{5}-493a^{4}-5119a^{3}+938a^{2}+5380a-1450\right){x}+20553a^{5}-11707a^{4}-115726a^{3}+25140a^{2}+124898a-33781$ |
| 47.1-a2 |
47.1-a |
$2$ |
$5$ |
6.6.1081856.1 |
$6$ |
$[6, 0]$ |
47.1 |
\( 47 \) |
\( - 47^{5} \) |
$128.10461$ |
$(2a^5-a^4-10a^3+8a+1)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$0.933502287$ |
$2993.115088$ |
3.22356 |
\( \frac{13560699203902695}{229345007} a^{5} - \frac{7929093749316729}{229345007} a^{4} - \frac{1624243186618611}{4879681} a^{3} + \frac{18191504505286714}{229345007} a^{2} + \frac{83672642355273096}{229345007} a - \frac{22714159122016115}{229345007} \) |
\( \bigl[a^{5} - a^{4} - 5 a^{3} + 2 a^{2} + 6 a\) , \( a^{5} - 2 a^{4} - 3 a^{3} + 5 a^{2} - 1\) , \( a^{4} - 5 a^{2} - a + 4\) , \( -3 a^{4} + a^{3} + 13 a^{2} - 5\) , \( 3 a^{5} - 3 a^{4} - 15 a^{3} + 10 a^{2} + 16 a - 7\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-5a^{3}+2a^{2}+6a\right){x}{y}+\left(a^{4}-5a^{2}-a+4\right){y}={x}^{3}+\left(a^{5}-2a^{4}-3a^{3}+5a^{2}-1\right){x}^{2}+\left(-3a^{4}+a^{3}+13a^{2}-5\right){x}+3a^{5}-3a^{4}-15a^{3}+10a^{2}+16a-7$ |
| 47.1-b1 |
47.1-b |
$2$ |
$2$ |
6.6.1081856.1 |
$6$ |
$[6, 0]$ |
47.1 |
\( 47 \) |
\( - 47^{2} \) |
$128.10461$ |
$(2a^5-a^4-10a^3+8a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.077325710$ |
$14592.51725$ |
3.25455 |
\( -\frac{122363909152}{2209} a^{5} - \frac{353980534185}{2209} a^{4} + \frac{13712106163}{47} a^{3} + \frac{1867875152277}{2209} a^{2} + \frac{371889180641}{2209} a - \frac{253512521526}{2209} \) |
\( \bigl[a^{5} - 6 a^{3} - a^{2} + 7 a\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 1\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 3\) , \( 43 a^{5} + 13 a^{4} - 250 a^{3} - 162 a^{2} + 238 a + 149\) , \( 1481 a^{5} + 402 a^{4} - 8776 a^{3} - 5351 a^{2} + 8915 a + 5407\bigr] \) |
${y}^2+\left(a^{5}-6a^{3}-a^{2}+7a\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+2a+3\right){y}={x}^{3}+\left(a^{4}-2a^{3}-3a^{2}+6a+1\right){x}^{2}+\left(43a^{5}+13a^{4}-250a^{3}-162a^{2}+238a+149\right){x}+1481a^{5}+402a^{4}-8776a^{3}-5351a^{2}+8915a+5407$ |
| 47.1-b2 |
47.1-b |
$2$ |
$2$ |
6.6.1081856.1 |
$6$ |
$[6, 0]$ |
47.1 |
\( 47 \) |
\( -47 \) |
$128.10461$ |
$(2a^5-a^4-10a^3+8a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.038662855$ |
$58370.06901$ |
3.25455 |
\( -\frac{112054}{47} a^{5} + \frac{135326}{47} a^{4} + 3564 a^{3} - \frac{236727}{47} a^{2} + \frac{990128}{47} a + \frac{591223}{47} \) |
\( \bigl[a^{4} - 5 a^{2} + 3\) , \( a^{5} - 2 a^{4} - 5 a^{3} + 7 a^{2} + 5 a - 2\) , \( a^{5} - a^{4} - 5 a^{3} + 2 a^{2} + 5 a + 1\) , \( 2 a^{5} - 6 a^{4} - 8 a^{3} + 21 a^{2} + 4 a - 5\) , \( 2 a^{5} + 3 a^{4} - 2 a^{3} - a^{2} - a - 1\bigr] \) |
${y}^2+\left(a^{4}-5a^{2}+3\right){x}{y}+\left(a^{5}-a^{4}-5a^{3}+2a^{2}+5a+1\right){y}={x}^{3}+\left(a^{5}-2a^{4}-5a^{3}+7a^{2}+5a-2\right){x}^{2}+\left(2a^{5}-6a^{4}-8a^{3}+21a^{2}+4a-5\right){x}+2a^{5}+3a^{4}-2a^{3}-a^{2}-a-1$ |
| 47.1-c1 |
47.1-c |
$4$ |
$6$ |
6.6.1081856.1 |
$6$ |
$[6, 0]$ |
47.1 |
\( 47 \) |
\( - 47^{3} \) |
$128.10461$ |
$(2a^5-a^4-10a^3+8a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$0.730692265$ |
$9848.125790$ |
3.45918 |
\( -\frac{17423948856}{103823} a^{5} + \frac{32224157618}{103823} a^{4} + \frac{983487568}{2209} a^{3} - \frac{58467426169}{103823} a^{2} - \frac{16161959950}{103823} a + \frac{8118919534}{103823} \) |
\( \bigl[a^{5} - 6 a^{3} - a^{2} + 8 a + 1\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 3\) , \( a^{5} - a^{4} - 4 a^{3} + a^{2} + 2 a + 2\) , \( 2 a^{5} - a^{4} - 3 a^{3} + 8 a^{2} - 2 a - 3\) , \( a^{5} + 5 a^{4} + 3 a^{3} - 5 a^{2} - 2 a - 4\bigr] \) |
${y}^2+\left(a^{5}-6a^{3}-a^{2}+8a+1\right){x}{y}+\left(a^{5}-a^{4}-4a^{3}+a^{2}+2a+2\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+3a+3\right){x}^{2}+\left(2a^{5}-a^{4}-3a^{3}+8a^{2}-2a-3\right){x}+a^{5}+5a^{4}+3a^{3}-5a^{2}-2a-4$ |
| 47.1-c2 |
47.1-c |
$4$ |
$6$ |
6.6.1081856.1 |
$6$ |
$[6, 0]$ |
47.1 |
\( 47 \) |
\( - 47^{6} \) |
$128.10461$ |
$(2a^5-a^4-10a^3+8a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.461384530$ |
$2462.031447$ |
3.45918 |
\( -\frac{13169624002415294460}{10779215329} a^{5} + \frac{11051641656481798944}{10779215329} a^{4} + \frac{1701450255542701371}{229345007} a^{3} - \frac{22980291315465018760}{10779215329} a^{2} - \frac{89116353692889766688}{10779215329} a + \frac{24473444901511420531}{10779215329} \) |
\( \bigl[2 a^{5} - a^{4} - 11 a^{3} + a^{2} + 12 a\) , \( -a^{3} + a^{2} + 4 a\) , \( 2 a^{5} - 2 a^{4} - 10 a^{3} + 5 a^{2} + 10 a - 2\) , \( a^{5} - 2 a^{4} + a^{3} - 6 a^{2} + 2 a - 2\) , \( 44 a^{5} - 77 a^{4} - 134 a^{3} + 152 a^{2} + 64 a - 34\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-11a^{3}+a^{2}+12a\right){x}{y}+\left(2a^{5}-2a^{4}-10a^{3}+5a^{2}+10a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a\right){x}^{2}+\left(a^{5}-2a^{4}+a^{3}-6a^{2}+2a-2\right){x}+44a^{5}-77a^{4}-134a^{3}+152a^{2}+64a-34$ |
| 47.1-c3 |
47.1-c |
$4$ |
$6$ |
6.6.1081856.1 |
$6$ |
$[6, 0]$ |
47.1 |
\( 47 \) |
\( -47 \) |
$128.10461$ |
$(2a^5-a^4-10a^3+8a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 1 \) |
$2.192076796$ |
$13.50908887$ |
3.45918 |
\( -\frac{28879258550583535046}{47} a^{5} + \frac{50245675129623334891}{47} a^{4} + 1826835480990573116 a^{3} - \frac{91641090704900686350}{47} a^{2} - \frac{42721984192460820482}{47} a + \frac{16595437813598028248}{47} \) |
\( \bigl[a^{5} - 6 a^{3} - a^{2} + 8 a + 1\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 3\) , \( a^{5} - a^{4} - 4 a^{3} + a^{2} + 2 a + 2\) , \( -8 a^{5} - 31 a^{4} + 47 a^{3} + 233 a^{2} + 8 a - 303\) , \( -115 a^{5} - 321 a^{4} + 732 a^{3} + 2198 a^{2} - 452 a - 2502\bigr] \) |
${y}^2+\left(a^{5}-6a^{3}-a^{2}+8a+1\right){x}{y}+\left(a^{5}-a^{4}-4a^{3}+a^{2}+2a+2\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+3a+3\right){x}^{2}+\left(-8a^{5}-31a^{4}+47a^{3}+233a^{2}+8a-303\right){x}-115a^{5}-321a^{4}+732a^{3}+2198a^{2}-452a-2502$ |
| 47.1-c4 |
47.1-c |
$4$ |
$6$ |
6.6.1081856.1 |
$6$ |
$[6, 0]$ |
47.1 |
\( 47 \) |
\( - 47^{2} \) |
$128.10461$ |
$(2a^5-a^4-10a^3+8a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 2 \) |
$4.384153592$ |
$3.377272218$ |
3.45918 |
\( -\frac{75821309689134022723951748286711}{2209} a^{5} + \frac{45077529317576597089391356811775}{2209} a^{4} + \frac{9109111067533549523397256431243}{47} a^{3} - \frac{102889549423672042160351921526305}{2209} a^{2} - \frac{469578941410894597000984454636639}{2209} a + \frac{127532965759371695390923008872004}{2209} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( a^{2} - a - 2\) , \( a^{5} - a^{4} - 4 a^{3} + 2 a^{2} + 3 a - 1\) , \( -835 a^{5} + 1228 a^{4} + 3259 a^{3} - 3308 a^{2} - 1155 a + 468\) , \( -23404 a^{5} + 32976 a^{4} + 94727 a^{3} - 87946 a^{2} - 42025 a + 16000\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{5}-a^{4}-4a^{3}+2a^{2}+3a-1\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-835a^{5}+1228a^{4}+3259a^{3}-3308a^{2}-1155a+468\right){x}-23404a^{5}+32976a^{4}+94727a^{3}-87946a^{2}-42025a+16000$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
