| 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 |
| 242.3-a1 |
242.3-a |
$2$ |
$2$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( - 2^{8} \cdot 11^{9} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$24.68670451$ |
5.554942512 |
\( \frac{8305}{256} a^{2} + \frac{9965}{128} a - \frac{46339}{128} \) |
\( \bigl[a\) , \( a^{2} + a - 4\) , \( a + 1\) , \( -5 a^{2} + 15 a - 3\) , \( 33 a^{2} - 93 a + 35\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(-5a^{2}+15a-3\right){x}+33a^{2}-93a+35$ |
| 242.3-a2 |
242.3-a |
$2$ |
$2$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{9} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$49.37340902$ |
5.554942512 |
\( -\frac{1064977}{16} a^{2} - \frac{1327117}{8} a + \frac{6682595}{8} \) |
\( \bigl[a\) , \( a^{2} - 2\) , \( a^{2} + a - 3\) , \( 14 a^{2} + 15 a - 111\) , \( -78 a^{2} - 26 a + 485\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(14a^{2}+15a-111\right){x}-78a^{2}-26a+485$ |
| 242.3-b1 |
242.3-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( - 2^{2} \cdot 11^{6} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$21.38345149$ |
1.202913127 |
\( \frac{427}{4} a^{2} + \frac{383}{2} a - \frac{47}{2} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{2} - 4\) , \( a^{2} - 3\) , \( -17089005789166286516 a^{2} + 9045494102123436959 a + 72613587255077018672\) , \( 239491187919164651072372415896 a^{2} - 126766656560365448750343743032 a - 1017631715112124311122237879909\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-17089005789166286516a^{2}+9045494102123436959a+72613587255077018672\right){x}+239491187919164651072372415896a^{2}-126766656560365448750343743032a-1017631715112124311122237879909$ |
| 242.3-b2 |
242.3-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( - 2 \cdot 11^{6} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$2.672931437$ |
1.202913127 |
\( -\frac{10154621765056003}{2} a^{2} + 2687505089603077 a + 21574207961612782 \) |
\( \bigl[a^{2} + a - 3\) , \( -a + 1\) , \( a\) , \( 163201859571 a^{2} - 86385450180 a - 693467637401\) , \( 52479892685555807 a^{2} - 27778477321846893 a - 222994439446883602\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(163201859571a^{2}-86385450180a-693467637401\right){x}+52479892685555807a^{2}-27778477321846893a-222994439446883602$ |
| 242.3-b3 |
242.3-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( 2^{2} \cdot 11^{6} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$10.69172574$ |
1.202913127 |
\( \frac{8525185037}{4} a^{2} - \frac{12150636535}{2} a + \frac{5069473275}{2} \) |
\( \bigl[a^{2} + a - 3\) , \( -a + 1\) , \( a\) , \( 10200132836 a^{2} - 5399099430 a - 43341797926\) , \( 820000263652323 a^{2} - 434039735261194 a - 3484296361562003\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(10200132836a^{2}-5399099430a-43341797926\right){x}+820000263652323a^{2}-434039735261194a-3484296361562003$ |
| 242.3-b4 |
242.3-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( 2 \cdot 11^{6} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$1.336465718$ |
1.202913127 |
\( \frac{89069256294443128323}{2} a^{2} - 125302919348047971845 a + 49111676741552886754 \) |
\( \bigl[1\) , \( a^{2} + a - 4\) , \( 1\) , \( 100 a^{2} + 76 a - 721\) , \( 1048 a^{2} + 485 a - 6877\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(100a^{2}+76a-721\right){x}+1048a^{2}+485a-6877$ |
| 242.3-b5 |
242.3-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( - 2^{4} \cdot 11^{6} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$10.69172574$ |
1.202913127 |
\( \frac{32251567931279}{16} a^{2} + \frac{21655756210331}{8} a - \frac{13765401110581}{8} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 20203900862562389334443 a^{2} - 10694259709833632274695 a - 85849214183464417082380\) , \( -2285799641956889659257421153575487 a^{2} + 1209911649340346379381685404090182 a + 9712683920681029332844506518986769\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(20203900862562389334443a^{2}-10694259709833632274695a-85849214183464417082380\right){x}-2285799641956889659257421153575487a^{2}+1209911649340346379381685404090182a+9712683920681029332844506518986769$ |
| 242.3-b6 |
242.3-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( 2^{8} \cdot 11^{6} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$42.76690299$ |
1.202913127 |
\( \frac{130050481}{256} a^{2} + \frac{87514925}{128} a - \frac{54929731}{128} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 1331292997660594700983 a^{2} - 704675456671176969100 a - 5656851044487640237345\) , \( -31621800609835111873402107774674 a^{2} + 16737943356313969364667113002303 a + 134365474859900078732316315405433\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(1331292997660594700983a^{2}-704675456671176969100a-5656851044487640237345\right){x}-31621800609835111873402107774674a^{2}+16737943356313969364667113002303a+134365474859900078732316315405433$ |
| 242.3-b7 |
242.3-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( 2^{16} \cdot 11^{6} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$21.38345149$ |
1.202913127 |
\( \frac{4108077233}{65536} a^{2} - \frac{5810733523}{32768} a + \frac{2294990397}{32768} \) |
\( \bigl[1\) , \( a + 1\) , \( a^{2} - 2\) , \( -26140776857754976733142982 a^{2} + 13836746608257972575168441 a + 111075834644485451677761735\) , \( -176958717384573892178034799698113749293 a^{2} + 93667183109988451030481895839197063871 a + 751922459614203146023948231356362753108\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-26140776857754976733142982a^{2}+13836746608257972575168441a+111075834644485451677761735\right){x}-176958717384573892178034799698113749293a^{2}+93667183109988451030481895839197063871a+751922459614203146023948231356362753108$ |
| 242.3-b8 |
242.3-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{6} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$42.76690299$ |
1.202913127 |
\( \frac{200481}{16} a^{2} - \frac{325027}{8} a + \frac{233973}{8} \) |
\( \bigl[1\) , \( -a^{2} + a + 3\) , \( 0\) , \( 41761256065998246 a^{2} - 22104925242738941 a - 177449446073241693\) , \( 6321701762221270525137686 a^{2} - 3346181557373420332949796 a - 26861799227819409800534166\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(41761256065998246a^{2}-22104925242738941a-177449446073241693\right){x}+6321701762221270525137686a^{2}-3346181557373420332949796a-26861799227819409800534166$ |
| 242.3-c1 |
242.3-c |
$2$ |
$2$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( - 2^{8} \cdot 11^{3} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.047891998$ |
$147.2051819$ |
1.189772066 |
\( \frac{8305}{256} a^{2} + \frac{9965}{128} a - \frac{46339}{128} \) |
\( \bigl[a\) , \( a^{2} - 3\) , \( 1\) , \( a^{2} - 2\) , \( -2 a^{2} + 2 a + 10\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(a^{2}-2\right){x}-2a^{2}+2a+10$ |
| 242.3-c2 |
242.3-c |
$2$ |
$2$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{3} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.023945999$ |
$294.4103639$ |
1.189772066 |
\( -\frac{1064977}{16} a^{2} - \frac{1327117}{8} a + \frac{6682595}{8} \) |
\( \bigl[a\) , \( -a^{2} + 2\) , \( a + 1\) , \( 422 a^{2} - 224 a - 1795\) , \( -6651 a^{2} + 3520 a + 28260\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(422a^{2}-224a-1795\right){x}-6651a^{2}+3520a+28260$ |
| 242.3-d1 |
242.3-d |
$4$ |
$4$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( - 2^{28} \cdot 11^{7} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 7 \) |
$0.167235986$ |
$15.23504241$ |
3.009879837 |
\( \frac{2181946399403}{2952790016} a^{2} + \frac{2000899833535}{1476395008} a + \frac{2482359162991}{1476395008} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{2} - a + 2\) , \( a^{2} + a - 2\) , \( 10351 a^{2} - 8483 a - 49431\) , \( 646151 a^{2} - 226218 a - 2535570\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(10351a^{2}-8483a-49431\right){x}+646151a^{2}-226218a-2535570$ |
| 242.3-d2 |
242.3-d |
$4$ |
$4$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( - 2^{7} \cdot 11^{10} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$0.668943946$ |
$3.808760603$ |
3.009879837 |
\( -\frac{2929687672739594901}{1874048} a^{2} + \frac{775365998006910655}{937024} a + \frac{6224327636401209327}{937024} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{2} + a + 4\) , \( 0\) , \( 185 a^{2} + 227 a - 211\) , \( -4950 a^{2} - 6821 a + 3934\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(185a^{2}+227a-211\right){x}-4950a^{2}-6821a+3934$ |
| 242.3-d3 |
242.3-d |
$4$ |
$4$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( 2^{14} \cdot 11^{8} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \cdot 7 \) |
$0.334471973$ |
$15.23504241$ |
3.009879837 |
\( -\frac{1196719965343}{1982464} a^{2} + \frac{318008627741}{991232} a + \frac{2546638361901}{991232} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{2} + a + 4\) , \( 0\) , \( -105 a^{2} - 138 a + 94\) , \( -1246 a^{2} - 1676 a + 1059\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(-105a^{2}-138a+94\right){x}-1246a^{2}-1676a+1059$ |
| 242.3-d4 |
242.3-d |
$4$ |
$4$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( - 2^{7} \cdot 11^{7} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$0.668943946$ |
$7.617521207$ |
3.009879837 |
\( \frac{7492235415}{1408} a^{2} + \frac{4985176571}{704} a - \frac{3177054421}{704} \) |
\( \bigl[1\) , \( a^{2} + a - 3\) , \( 1\) , \( 379 a^{2} - 202 a - 1613\) , \( 9540 a^{2} - 5059 a - 40554\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(379a^{2}-202a-1613\right){x}+9540a^{2}-5059a-40554$ |
| 242.3-e1 |
242.3-e |
$2$ |
$2$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( - 2^{2} \cdot 11^{9} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.854761088$ |
$18.08166434$ |
2.608319931 |
\( -\frac{3565263}{4} a^{2} + \frac{1011171}{2} a + \frac{7697231}{2} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 479825 a^{2} - 254167 a - 2039183\) , \( 264702823 a^{2} - 140109753 a - 1124756159\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(479825a^{2}-254167a-2039183\right){x}+264702823a^{2}-140109753a-1124756159$ |
| 242.3-e2 |
242.3-e |
$2$ |
$2$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( - 2 \cdot 11^{9} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.709522177$ |
$18.08166434$ |
2.608319931 |
\( \frac{22375}{2} a^{2} - 29801 a + 10226 \) |
\( \bigl[1\) , \( -a^{2} + a + 3\) , \( 0\) , \( -a^{2} - 5 a - 5\) , \( 7 a^{2} - 6 a - 34\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-a^{2}-5a-5\right){x}+7a^{2}-6a-34$ |
| 242.3-f1 |
242.3-f |
$2$ |
$2$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( - 2 \cdot 11^{3} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$63.60912793$ |
1.789146505 |
\( \frac{22375}{2} a^{2} - 29801 a + 10226 \) |
\( \bigl[a^{2} + a - 3\) , \( -a - 1\) , \( a^{2} - 3\) , \( -a - 3\) , \( -2\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a-3\right){x}-2$ |
| 242.3-f2 |
242.3-f |
$2$ |
$2$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( - 2^{2} \cdot 11^{3} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$31.80456396$ |
1.789146505 |
\( -\frac{3565263}{4} a^{2} + \frac{1011171}{2} a + \frac{7697231}{2} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{2} + 2\) , \( a\) , \( 11155 a^{2} - 5917 a - 47423\) , \( 950125 a^{2} - 502958 a - 4037290\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(11155a^{2}-5917a-47423\right){x}+950125a^{2}-502958a-4037290$ |
*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.
