| 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 |
| 2.1-a1 |
2.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( 2^{8} \) |
$1.41378$ |
$(a+6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$8.806287547$ |
1.323842211 |
\( \frac{8987}{256} a - \frac{64527}{256} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 10 a - 72\) , \( 97 a - 694\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(10a-72\right){x}+97a-694$ |
| 2.1-b1 |
2.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( 2^{8} \) |
$1.41378$ |
$(a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$0.052774385$ |
$34.83900471$ |
2.211170698 |
\( \frac{8987}{256} a - \frac{64527}{256} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -187279 a - 1152153\) , \( 4774539330 a + 29373287513\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-187279a-1152153\right){x}+4774539330a+29373287513$ |
| 2.2-a1 |
2.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( 2^{8} \) |
$1.41378$ |
$(a-7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$8.806287547$ |
1.323842211 |
\( -\frac{8987}{256} a - \frac{13885}{64} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -10 a - 62\) , \( -97 a - 597\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-10a-62\right){x}-97a-597$ |
| 2.2-b1 |
2.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( 2^{8} \) |
$1.41378$ |
$(a-7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$0.052774385$ |
$34.83900471$ |
2.211170698 |
\( -\frac{8987}{256} a - \frac{13885}{64} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 187279 a - 1339432\) , \( -4774539330 a + 34147826843\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(187279a-1339432\right){x}-4774539330a+34147826843$ |
| 4.1-a1 |
4.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{40} \) |
$1.68128$ |
$(a+6), (a-7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2^{3} \) |
$3.344674174$ |
$0.770926695$ |
3.100989179 |
\( -\frac{1130063023676147}{68719476736} a - \frac{1751035722404965}{17179869184} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -872614 a - 5368377\) , \( -1148958419 a - 7068469583\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-872614a-5368377\right){x}-1148958419a-7068469583$ |
| 4.1-a2 |
4.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{40} \) |
$1.68128$ |
$(a+6), (a-7)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$0.371630463$ |
$6.938340258$ |
3.100989179 |
\( \frac{1130063023676147}{68719476736} a - \frac{8134205913296007}{68719476736} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -248384 a - 1528072\) , \( 315070330 a + 1938333880\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-248384a-1528072\right){x}+315070330a+1938333880$ |
| 4.1-a3 |
4.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{24} \) |
$1.68128$ |
$(a+6), (a-7)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1.114891391$ |
$6.938340258$ |
3.100989179 |
\( \frac{2352637}{4096} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( 26001 a + 159963\) , \( -8339907 a - 51307679\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(26001a+159963\right){x}-8339907a-51307679$ |
| 4.1-b1 |
4.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{10} \) |
$1.68128$ |
$(a+6), (a-7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$6.613783411$ |
$2.014999682$ |
2.003402969 |
\( -\frac{8613448793125}{512} a - \frac{52990516263375}{512} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( -577018 a - 3549851\) , \( -615401917 a - 3785994046\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-577018a-3549851\right){x}-615401917a-3785994046$ |
| 4.1-b2 |
4.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{6} \) |
$1.68128$ |
$(a+6), (a-7)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$2.204594470$ |
$18.13499714$ |
2.003402969 |
\( -\frac{42875}{8} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( -6843 a - 42096\) , \( -914367 a - 5625254\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-6843a-42096\right){x}-914367a-5625254$ |
| 4.1-b3 |
4.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{10} \) |
$1.68128$ |
$(a+6), (a-7)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$0.734864823$ |
$18.13499714$ |
2.003402969 |
\( \frac{8613448793125}{512} a - \frac{15400991264125}{128} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( 47212 a + 290454\) , \( 4411418 a + 27139334\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(47212a+290454\right){x}+4411418a+27139334$ |
| 4.1-c1 |
4.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{10} \) |
$1.68128$ |
$(a+6), (a-7)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$0.734864823$ |
$18.13499714$ |
2.003402969 |
\( -\frac{8613448793125}{512} a - \frac{52990516263375}{512} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -47213 a + 337667\) , \( -4411419 a + 31550753\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-47213a+337667\right){x}-4411419a+31550753$ |
| 4.1-c2 |
4.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{6} \) |
$1.68128$ |
$(a+6), (a-7)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$2.204594470$ |
$18.13499714$ |
2.003402969 |
\( -\frac{42875}{8} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( 6842 a - 48938\) , \( 914366 a - 6539620\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(6842a-48938\right){x}+914366a-6539620$ |
| 4.1-c3 |
4.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{10} \) |
$1.68128$ |
$(a+6), (a-7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$6.613783411$ |
$2.014999682$ |
2.003402969 |
\( \frac{8613448793125}{512} a - \frac{15400991264125}{128} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( 577017 a - 4126868\) , \( 615401916 a - 4401395962\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(577017a-4126868\right){x}+615401916a-4401395962$ |
| 4.1-d1 |
4.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{40} \) |
$1.68128$ |
$(a+6), (a-7)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$0.371630463$ |
$6.938340258$ |
3.100989179 |
\( -\frac{1130063023676147}{68719476736} a - \frac{1751035722404965}{17179869184} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( 248383 a - 1776456\) , \( -315070331 a + 2253404210\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(248383a-1776456\right){x}-315070331a+2253404210$ |
| 4.1-d2 |
4.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{40} \) |
$1.68128$ |
$(a+6), (a-7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2^{3} \) |
$3.344674174$ |
$0.770926695$ |
3.100989179 |
\( \frac{1130063023676147}{68719476736} a - \frac{8134205913296007}{68719476736} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( 872613 a - 6240991\) , \( 1148958418 a - 8217428002\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(872613a-6240991\right){x}+1148958418a-8217428002$ |
| 4.1-d3 |
4.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{24} \) |
$1.68128$ |
$(a+6), (a-7)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1.114891391$ |
$6.938340258$ |
3.100989179 |
\( \frac{2352637}{4096} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( -26002 a + 185964\) , \( 8339906 a - 59647586\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-26002a+185964\right){x}+8339906a-59647586$ |
| 6.1-a1 |
6.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3^{10} \) |
$1.86064$ |
$(a-7), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$25.70567700$ |
0.966078500 |
\( -\frac{27253633}{648} a + \frac{136079617}{486} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -255007 a - 1568804\) , \( 188733126 a + 1161098905\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-255007a-1568804\right){x}+188733126a+1161098905$ |
| 6.1-a2 |
6.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{6} \cdot 3^{5} \) |
$1.86064$ |
$(a-7), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$25.70567700$ |
0.966078500 |
\( \frac{60250704371}{1728} a + \frac{92667370229}{432} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -7716002993 a - 47469370050\) , \( 951528250845569 a + 5853865882560073\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-7716002993a-47469370050\right){x}+951528250845569a+5853865882560073$ |
| 6.1-b1 |
6.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3^{10} \) |
$1.86064$ |
$(a-7), (-46a-283)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$2.511646627$ |
$3.853936275$ |
2.182718290 |
\( -\frac{27253633}{648} a + \frac{136079617}{486} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( 26122 a - 186814\) , \( 5973564 a - 42723357\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(26122a-186814\right){x}+5973564a-42723357$ |
| 6.1-b2 |
6.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{6} \cdot 3^{5} \) |
$1.86064$ |
$(a-7), (-46a-283)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1.255823313$ |
$7.707872550$ |
2.182718290 |
\( \frac{60250704371}{1728} a + \frac{92667370229}{432} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 3\) , \( -3 a - 6\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+3{x}-3a-6$ |
| 6.1-c1 |
6.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{10} \cdot 3^{3} \) |
$1.86064$ |
$(a-7), (-46a-283)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3 \cdot 5 \) |
$1$ |
$3.204184574$ |
7.225237825 |
\( -\frac{1264043833}{9216} a - \frac{1944121807}{2304} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -6 a + 18\) , \( -9 a + 45\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a+18\right){x}-9a+45$ |
| 6.1-d1 |
6.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{10} \cdot 3^{3} \) |
$1.86064$ |
$(a-7), (-46a-283)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.056828548$ |
$26.12185764$ |
1.338952701 |
\( -\frac{1264043833}{9216} a - \frac{1944121807}{2304} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 503558533 a - 3601484538\) , \( -950571623996724 a + 6798552273755990\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(503558533a-3601484538\right){x}-950571623996724a+6798552273755990$ |
| 6.1-e1 |
6.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3 \) |
$1.86064$ |
$(a-7), (-46a-283)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$6.011016291$ |
$6.743827619$ |
4.570450309 |
\( -\frac{664325497}{24} a + \frac{1187823671}{6} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -32 a - 181\) , \( -775 a - 4780\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-32a-181\right){x}-775a-4780$ |
| 6.1-e2 |
6.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{12} \cdot 3 \) |
$1.86064$ |
$(a-7), (-46a-283)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.502754072$ |
$26.97531047$ |
4.570450309 |
\( -\frac{1847989}{12288} a + \frac{8865485}{3072} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( 775751429 a - 5548226450\) , \( -22424537916272 a + 160381805421021\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(775751429a-5548226450\right){x}-22424537916272a+160381805421021$ |
| 6.1-e3 |
6.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{6} \cdot 3^{2} \) |
$1.86064$ |
$(a-7), (-46a-283)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$3.005508145$ |
$13.48765523$ |
4.570450309 |
\( \frac{2791217}{64} a + \frac{13836149}{48} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -1373836 a - 8451928\) , \( -2258018730 a - 13891483309\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-1373836a-8451928\right){x}-2258018730a-13891483309$ |
| 6.1-e4 |
6.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3^{4} \) |
$1.86064$ |
$(a-7), (-46a-283)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$6.011016291$ |
$3.371913809$ |
4.570450309 |
\( \frac{648189313321}{24} a + \frac{2990778232775}{18} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -21972571 a - 135176733\) , \( -144607302719 a - 889633865324\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-21972571a-135176733\right){x}-144607302719a-889633865324$ |
| 6.1-f1 |
6.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3 \) |
$1.86064$ |
$(a-7), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$44.82516825$ |
0.842316491 |
\( -\frac{664325497}{24} a + \frac{1187823671}{6} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( 15647397 a - 111911224\) , \( -86932757108 a + 621748933578\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(15647397a-111911224\right){x}-86932757108a+621748933578$ |
| 6.1-f2 |
6.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{12} \cdot 3 \) |
$1.86064$ |
$(a-7), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$22.41258412$ |
0.842316491 |
\( -\frac{1847989}{12288} a + \frac{8865485}{3072} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 13\) , \( -a + 2\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+13{x}-a+2$ |
| 6.1-f3 |
6.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{6} \cdot 3^{2} \) |
$1.86064$ |
$(a-7), (-46a-283)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$44.82516825$ |
0.842316491 |
\( \frac{2791217}{64} a + \frac{13836149}{48} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 529 a - 3787\) , \( -16128 a + 115332\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(529a-3787\right){x}-16128a+115332$ |
| 6.1-f4 |
6.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3^{4} \) |
$1.86064$ |
$(a-7), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$22.41258412$ |
0.842316491 |
\( \frac{648189313321}{24} a + \frac{2990778232775}{18} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -326 a + 2328\) , \( -64212 a + 459232\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-326a+2328\right){x}-64212a+459232$ |
| 6.2-a1 |
6.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{6} \cdot 3^{5} \) |
$1.86064$ |
$(a+6), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$25.70567700$ |
0.966078500 |
\( -\frac{60250704371}{1728} a + \frac{430920185287}{1728} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( 7716002992 a - 55185373043\) , \( -951528250845570 a + 6805394133405642\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(7716002992a-55185373043\right){x}-951528250845570a+6805394133405642$ |
| 6.2-a2 |
6.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3^{10} \) |
$1.86064$ |
$(a+6), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$25.70567700$ |
0.966078500 |
\( \frac{27253633}{648} a + \frac{462557569}{1944} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( 255006 a - 1823810\) , \( -188733127 a + 1349832032\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(255006a-1823810\right){x}-188733127a+1349832032$ |
| 6.2-b1 |
6.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{6} \cdot 3^{5} \) |
$1.86064$ |
$(a+6), (-46a-283)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1.255823313$ |
$7.707872550$ |
2.182718290 |
\( -\frac{60250704371}{1728} a + \frac{430920185287}{1728} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -a + 4\) , \( 2 a - 8\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-a+4\right){x}+2a-8$ |
| 6.2-b2 |
6.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3^{10} \) |
$1.86064$ |
$(a+6), (-46a-283)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$2.511646627$ |
$3.853936275$ |
2.182718290 |
\( \frac{27253633}{648} a + \frac{462557569}{1944} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -26123 a - 160692\) , \( -5973565 a - 36749793\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-26123a-160692\right){x}-5973565a-36749793$ |
| 6.2-c1 |
6.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{10} \cdot 3^{3} \) |
$1.86064$ |
$(a+6), (-46a-283)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3 \cdot 5 \) |
$1$ |
$3.204184574$ |
7.225237825 |
\( \frac{1264043833}{9216} a - \frac{9040531061}{9216} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 6 a + 56\) , \( 14 a + 93\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(6a+56\right){x}+14a+93$ |
| 6.2-d1 |
6.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{10} \cdot 3^{3} \) |
$1.86064$ |
$(a+6), (-46a-283)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.056828548$ |
$26.12185764$ |
1.338952701 |
\( \frac{1264043833}{9216} a - \frac{9040531061}{9216} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( -503558533 a - 3097926005\) , \( 950571120438191 a + 5847977551833261\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-503558533a-3097926005\right){x}+950571120438191a+5847977551833261$ |
| 6.2-e1 |
6.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3^{4} \) |
$1.86064$ |
$(a+6), (-46a-283)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$6.011016291$ |
$3.371913809$ |
4.570450309 |
\( -\frac{648189313321}{24} a + \frac{13907680871063}{72} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 21972570 a - 157149304\) , \( 144607302718 a - 1034241168043\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(21972570a-157149304\right){x}+144607302718a-1034241168043$ |
| 6.2-e2 |
6.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{12} \cdot 3 \) |
$1.86064$ |
$(a+6), (-46a-283)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.502754072$ |
$26.97531047$ |
4.570450309 |
\( \frac{1847989}{12288} a + \frac{33613951}{12288} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -775751429 a - 4772475021\) , \( 22424537916272 a + 137957267504749\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-775751429a-4772475021\right){x}+22424537916272a+137957267504749$ |
| 6.2-e3 |
6.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{6} \cdot 3^{2} \) |
$1.86064$ |
$(a+6), (-46a-283)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$3.005508145$ |
$13.48765523$ |
4.570450309 |
\( -\frac{2791217}{64} a + \frac{63718247}{192} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 1373835 a - 9825764\) , \( 2258018729 a - 16149502039\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(1373835a-9825764\right){x}+2258018729a-16149502039$ |
| 6.2-e4 |
6.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3 \) |
$1.86064$ |
$(a+6), (-46a-283)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$6.011016291$ |
$6.743827619$ |
4.570450309 |
\( \frac{664325497}{24} a + \frac{4086969187}{24} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 32 a - 213\) , \( 775 a - 5555\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(32a-213\right){x}+775a-5555$ |
| 6.2-f1 |
6.2-f |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3^{4} \) |
$1.86064$ |
$(a+6), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$22.41258412$ |
0.842316491 |
\( -\frac{648189313321}{24} a + \frac{13907680871063}{72} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( 325 a + 2003\) , \( 64211 a + 395021\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(325a+2003\right){x}+64211a+395021$ |
| 6.2-f2 |
6.2-f |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{12} \cdot 3 \) |
$1.86064$ |
$(a+6), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$22.41258412$ |
0.842316491 |
\( \frac{1847989}{12288} a + \frac{33613951}{12288} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 13\) , \( a + 1\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+13{x}+a+1$ |
| 6.2-f3 |
6.2-f |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{6} \cdot 3^{2} \) |
$1.86064$ |
$(a+6), (-46a-283)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$44.82516825$ |
0.842316491 |
\( -\frac{2791217}{64} a + \frac{63718247}{192} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -530 a - 3257\) , \( 16127 a + 99205\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-530a-3257\right){x}+16127a+99205$ |
| 6.2-f4 |
6.2-f |
$4$ |
$4$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3 \) |
$1.86064$ |
$(a+6), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$44.82516825$ |
0.842316491 |
\( \frac{664325497}{24} a + \frac{4086969187}{24} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -15647397 a - 96263827\) , \( 86932757108 a + 534816176470\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-15647397a-96263827\right){x}+86932757108a+534816176470$ |
| 12.2-a1 |
12.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{6} \) |
$2.21269$ |
$(a+6), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$11.16776518$ |
3.777392854 |
\( -\frac{2641}{27} a + \frac{27172}{27} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 11 a + 55\) , \( 29 a + 172\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(11a+55\right){x}+29a+172$ |
| 12.2-a2 |
12.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{3} \) |
$2.21269$ |
$(a+6), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3^{2} \) |
$1$ |
$22.33553036$ |
3.777392854 |
\( \frac{3337}{9} a + \frac{37228}{9} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -17494 a - 107572\) , \( -2830431 a - 17412896\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-17494a-107572\right){x}-2830431a-17412896$ |
| 12.2-b1 |
12.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{6} \) |
$2.21269$ |
$(a+6), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$14.41190537$ |
1.624897714 |
\( -\frac{2641}{27} a + \frac{27172}{27} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -69630246 a + 498000355\) , \( 222602880028 a - 1592070789440\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-69630246a+498000355\right){x}+222602880028a-1592070789440$ |
| 12.2-b2 |
12.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{3} \) |
$2.21269$ |
$(a+6), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1$ |
$28.82381074$ |
1.624897714 |
\( \frac{3337}{9} a + \frac{37228}{9} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 9681 a - 69232\) , \( 397848 a - 2845432\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(9681a-69232\right){x}+397848a-2845432$ |
| 12.3-a1 |
12.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
12.3 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{6} \) |
$2.21269$ |
$(a-7), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$11.16776518$ |
3.777392854 |
\( \frac{2641}{27} a + \frac{8177}{9} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 12 a + 87\) , \( 36 a + 180\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(12a+87\right){x}+36a+180$ |
| 12.3-a2 |
12.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{177}) \) |
$2$ |
$[2, 0]$ |
12.3 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{3} \) |
$2.21269$ |
$(a-7), (-46a-283)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3^{2} \) |
$1$ |
$22.33553036$ |
3.777392854 |
\( -\frac{3337}{9} a + \frac{40565}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 17495 a - 125022\) , \( 2847925 a - 20368349\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(17495a-125022\right){x}+2847925a-20368349$ |
*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.
