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 |
726.1-a1 |
726.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{2} \cdot 3^{8} \cdot 11^{8} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.195042991$ |
$4.813898015$ |
1.533245153 |
\( -\frac{192100033}{2371842} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( 239 a - 589\) , \( -14693 a + 35989\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(239a-589\right){x}-14693a+35989$ |
726.1-a2 |
726.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 11^{2} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.195042991$ |
$19.25559206$ |
1.533245153 |
\( \frac{912673}{528} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( 39 a - 99\) , \( 57 a - 141\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(39a-99\right){x}+57a-141$ |
726.1-a3 |
726.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{4} \cdot 3^{4} \cdot 11^{4} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.390085982$ |
$19.25559206$ |
1.533245153 |
\( \frac{1180932193}{4356} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( -440 a - 1079\) , \( 7267 a + 17799\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-440a-1079\right){x}+7267a+17799$ |
726.1-a4 |
726.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.780171965$ |
$19.25559206$ |
1.533245153 |
\( \frac{4824238966273}{66} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( 7039 a - 17249\) , \( -494017 a + 1210089\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(7039a-17249\right){x}-494017a+1210089$ |
726.1-b1 |
726.1-b |
$4$ |
$10$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{2} \cdot 3^{4} \cdot 11^{20} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$1$ |
\( 2^{4} \cdot 5 \) |
$8.015579573$ |
$0.056797834$ |
3.717244094 |
\( -\frac{112427521449300721}{466873642818} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -10055\) , \( -390309\bigr] \) |
${y}^2+{x}{y}={x}^{3}-10055{x}-390309$ |
726.1-b2 |
726.1-b |
$4$ |
$10$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{10} \cdot 3^{20} \cdot 11^{4} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
$1$ |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2^{4} \cdot 5^{2} \) |
$1.603115914$ |
$1.419945868$ |
3.717244094 |
\( \frac{168105213359}{228637728} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 115\) , \( 561\bigr] \) |
${y}^2+{x}{y}={x}^{3}+115{x}+561$ |
726.1-b3 |
726.1-b |
$4$ |
$10$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{20} \cdot 3^{10} \cdot 11^{2} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
$1$ |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2^{3} \cdot 5^{2} \) |
$0.801557957$ |
$5.679783475$ |
3.717244094 |
\( \frac{10091699281}{2737152} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -45\) , \( 81\bigr] \) |
${y}^2+{x}{y}={x}^{3}-45{x}+81$ |
726.1-b4 |
726.1-b |
$4$ |
$10$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{4} \cdot 3^{2} \cdot 11^{10} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$1$ |
\( 2^{3} \cdot 5 \) |
$4.007789786$ |
$0.227191339$ |
3.717244094 |
\( \frac{112763292123580561}{1932612} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -10065\) , \( -389499\bigr] \) |
${y}^2+{x}{y}={x}^{3}-10065{x}-389499$ |
726.1-c1 |
726.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{2} \cdot 3^{8} \cdot 11^{8} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$1.214828373$ |
3.967612853 |
\( -\frac{192100033}{2371842} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -12\) , \( -81\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-12{x}-81$ |
726.1-c2 |
726.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 11^{2} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$19.43725397$ |
3.967612853 |
\( \frac{912673}{528} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -2\) , \( -1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-2{x}-1$ |
726.1-c3 |
726.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{4} \cdot 3^{4} \cdot 11^{4} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{4} \) |
$1$ |
$4.859313493$ |
3.967612853 |
\( \frac{1180932193}{4356} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -22\) , \( -49\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-22{x}-49$ |
726.1-c4 |
726.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$1.214828373$ |
3.967612853 |
\( \frac{4824238966273}{66} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -352\) , \( -2689\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-352{x}-2689$ |
726.1-d1 |
726.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{6} \cdot 3^{4} \cdot 11^{12} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{4} \cdot 3^{2} \) |
$1.548328844$ |
$2.444595345$ |
6.180940330 |
\( -\frac{7357983625}{127552392} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 811 a - 1984\) , \( -109228 a + 267554\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(811a-1984\right){x}-109228a+267554$ |
726.1-d2 |
726.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{2} \cdot 3^{12} \cdot 11^{4} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.516109614$ |
$2.444595345$ |
6.180940330 |
\( \frac{9938375}{176418} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 89 a + 221\) , \( -3920 a - 9601\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(89a+221\right){x}-3920a-9601$ |
726.1-d3 |
726.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{4} \cdot 3^{6} \cdot 11^{2} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.258054807$ |
$9.778381380$ |
6.180940330 |
\( \frac{18609625}{1188} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -111 a - 269\) , \( -952 a - 2331\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-111a-269\right){x}-952a-2331$ |
726.1-d4 |
726.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{12} \cdot 3^{2} \cdot 11^{6} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$0.774164422$ |
$9.778381380$ |
6.180940330 |
\( \frac{57736239625}{255552} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 1611 a - 3944\) , \( -54572 a + 133674\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(1611a-3944\right){x}-54572a+133674$ |
726.1-e1 |
726.1-e |
$4$ |
$10$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{2} \cdot 3^{4} \cdot 11^{20} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.884125736$ |
3.609428203 |
\( -\frac{112427521449300721}{466873642818} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 201100 a - 492698\) , \( -77080082 a + 188807168\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(201100a-492698\right){x}-77080082a+188807168$ |
726.1-e2 |
726.1-e |
$4$ |
$10$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{10} \cdot 3^{20} \cdot 11^{4} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.884125736$ |
3.609428203 |
\( \frac{168105213359}{228637728} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 2298 a + 5632\) , \( -108779 a - 266452\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(2298a+5632\right){x}-108779a-266452$ |
726.1-e3 |
726.1-e |
$4$ |
$10$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{20} \cdot 3^{10} \cdot 11^{2} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$3.536502944$ |
3.609428203 |
\( \frac{10091699281}{2737152} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -902 a - 2208\) , \( -16939 a - 41492\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-902a-2208\right){x}-16939a-41492$ |
726.1-e4 |
726.1-e |
$4$ |
$10$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{4} \cdot 3^{2} \cdot 11^{10} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$3.536502944$ |
3.609428203 |
\( \frac{112763292123580561}{1932612} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -201302 a - 493188\) , \( 76919501 a + 188413828\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-201302a-493188\right){x}+76919501a+188413828$ |
726.1-f1 |
726.1-f |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{6} \cdot 3^{4} \cdot 11^{12} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$0.635354791$ |
1.556295044 |
\( -\frac{7357983625}{127552392} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -41\) , \( -556\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-41{x}-556$ |
726.1-f2 |
726.1-f |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{2} \cdot 3^{12} \cdot 11^{4} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$5.718193122$ |
1.556295044 |
\( \frac{9938375}{176418} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 4\) , \( 20\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+4{x}+20$ |
726.1-f3 |
726.1-f |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{4} \cdot 3^{6} \cdot 11^{2} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$22.87277248$ |
1.556295044 |
\( \frac{18609625}{1188} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -6\) , \( 4\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-6{x}+4$ |
726.1-f4 |
726.1-f |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
726.1 |
\( 2 \cdot 3 \cdot 11^{2} \) |
\( 2^{12} \cdot 3^{2} \cdot 11^{6} \) |
$2.27237$ |
$(-a+2), (a+3), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$2.541419165$ |
1.556295044 |
\( \frac{57736239625}{255552} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -81\) , \( -284\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-81{x}-284$ |
*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.