| 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 |
| 46818.2-a1 |
46818.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{12} \cdot 3^{12} \cdot 17^{2} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$0.562629009$ |
$1.396783906$ |
3.143484585 |
\( \frac{3048625}{1088} \) |
\( \bigl[i\) , \( 1\) , \( 0\) , \( -27\) , \( 27\bigr] \) |
${y}^2+i{x}{y}={x}^{3}+{x}^{2}-27{x}+27$ |
| 46818.2-a2 |
46818.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{2} \cdot 3^{12} \cdot 17^{12} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{5} \cdot 3^{2} \) |
$0.843943514$ |
$0.232797317$ |
3.143484585 |
\( \frac{159661140625}{48275138} \) |
\( \bigl[i\) , \( 1\) , \( 0\) , \( -1017\) , \( -8883\bigr] \) |
${y}^2+i{x}{y}={x}^{3}+{x}^{2}-1017{x}-8883$ |
| 46818.2-a3 |
46818.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{6} \cdot 3^{12} \cdot 17^{4} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{5} \) |
$0.281314504$ |
$0.698391953$ |
3.143484585 |
\( \frac{8805624625}{2312} \) |
\( \bigl[i\) , \( 1\) , \( 0\) , \( -387\) , \( 2835\bigr] \) |
${y}^2+i{x}{y}={x}^{3}+{x}^{2}-387{x}+2835$ |
| 46818.2-a4 |
46818.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{4} \cdot 3^{12} \cdot 17^{6} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1.687887029$ |
$0.465594635$ |
3.143484585 |
\( \frac{120920208625}{19652} \) |
\( \bigl[i\) , \( 1\) , \( 0\) , \( -927\) , \( -11097\bigr] \) |
${y}^2+i{x}{y}={x}^{3}+{x}^{2}-927{x}-11097$ |
| 46818.2-b1 |
46818.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{2} \cdot 3^{16} \cdot 17^{16} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$4.017174891$ |
$0.061251382$ |
3.936920247 |
\( -\frac{491411892194497}{125563633938} \) |
\( \bigl[i\) , \( 1\) , \( 0\) , \( -14796\) , \( -835434\bigr] \) |
${y}^2+i{x}{y}={x}^{3}+{x}^{2}-14796{x}-835434$ |
| 46818.2-b2 |
46818.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{4} \cdot 3^{44} \cdot 17^{2} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$8.034349782$ |
$0.122502764$ |
3.936920247 |
\( \frac{1276229915423}{2927177028} \) |
\( \bigl[i\) , \( 1\) , \( 0\) , \( 2034\) , \( -60264\bigr] \) |
${y}^2+i{x}{y}={x}^{3}+{x}^{2}+2034{x}-60264$ |
| 46818.2-b3 |
46818.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{8} \cdot 3^{28} \cdot 17^{4} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$4.017174891$ |
$0.245005529$ |
3.936920247 |
\( \frac{163936758817}{30338064} \) |
\( \bigl[i\) , \( 1\) , \( 0\) , \( -1026\) , \( -10692\bigr] \) |
${y}^2+i{x}{y}={x}^{3}+{x}^{2}-1026{x}-10692$ |
| 46818.2-b4 |
46818.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{16} \cdot 3^{20} \cdot 17^{2} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.008587445$ |
$0.490011059$ |
3.936920247 |
\( \frac{4354703137}{352512} \) |
\( \bigl[i\) , \( 1\) , \( 0\) , \( -306\) , \( 1836\bigr] \) |
${y}^2+i{x}{y}={x}^{3}+{x}^{2}-306{x}+1836$ |
| 46818.2-b5 |
46818.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{4} \cdot 3^{20} \cdot 17^{8} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$8.034349782$ |
$0.122502764$ |
3.936920247 |
\( \frac{576615941610337}{27060804} \) |
\( \bigl[i\) , \( 1\) , \( 0\) , \( -15606\) , \( -754272\bigr] \) |
${y}^2+i{x}{y}={x}^{3}+{x}^{2}-15606{x}-754272$ |
| 46818.2-b6 |
46818.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{2} \cdot 3^{16} \cdot 17^{4} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$4.017174891$ |
$0.061251382$ |
3.936920247 |
\( \frac{2361739090258884097}{5202} \) |
\( \bigl[i\) , \( 1\) , \( 0\) , \( -249696\) , \( -48087270\bigr] \) |
${y}^2+i{x}{y}={x}^{3}+{x}^{2}-249696{x}-48087270$ |
| 46818.2-c1 |
46818.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2 \cdot 3^{14} \cdot 17^{7} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.169728025$ |
2.715648406 |
\( \frac{1979660058649925}{501126} a - \frac{547309863864799}{167042} \) |
\( \bigl[i\) , \( 1\) , \( 1\) , \( -14255 i + 723\) , \( 423947 i - 497800\bigr] \) |
${y}^2+i{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-14255i+723\right){x}+423947i-497800$ |
| 46818.2-c2 |
46818.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{2} \cdot 3^{16} \cdot 17^{8} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{5} \) |
$1$ |
$0.339456050$ |
2.715648406 |
\( -\frac{9390341072075}{144825414} a - \frac{6456168132412}{217238121} \) |
\( \bigl[i\) , \( 1\) , \( 1\) , \( -890 i + 48\) , \( 6527 i - 7750\bigr] \) |
${y}^2+i{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-890i+48\right){x}+6527i-7750$ |
| 46818.2-c3 |
46818.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2 \cdot 3^{14} \cdot 17^{13} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.169728025$ |
2.715648406 |
\( -\frac{461275687224792005}{3495733423378566} a + \frac{140679328163848447}{1165244474459522} \) |
\( \bigl[1\) , \( -1\) , \( i\) , \( -485 i - 627\) , \( -3179 i + 25732\bigr] \) |
${y}^2+{x}{y}+i{y}={x}^{3}-{x}^{2}+\left(-485i-627\right){x}-3179i+25732$ |
| 46818.2-c4 |
46818.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{4} \cdot 3^{20} \cdot 17^{4} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.678912101$ |
2.715648406 |
\( \frac{88739980}{132651} a + \frac{1762314767}{1591812} \) |
\( \bigl[1\) , \( -1\) , \( i\) , \( -80 i + 48\) , \( -209 i - 188\bigr] \) |
${y}^2+{x}{y}+i{y}={x}^{3}-{x}^{2}+\left(-80i+48\right){x}-209i-188$ |
| 46818.2-d1 |
46818.2-d |
$4$ |
$6$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{6} \cdot 3^{36} \cdot 17^{4} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{5} \cdot 3 \) |
$1$ |
$0.138622644$ |
3.326943479 |
\( -\frac{1107111813625}{1228691592} \) |
\( \bigl[i\) , \( 1\) , \( i\) , \( -1939\) , \( 55681\bigr] \) |
${y}^2+i{x}{y}+i{y}={x}^{3}+{x}^{2}-1939{x}+55681$ |
| 46818.2-d2 |
46818.2-d |
$4$ |
$6$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{18} \cdot 3^{20} \cdot 17^{12} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{5} \cdot 3^{4} \) |
$1$ |
$0.046207548$ |
3.326943479 |
\( \frac{655215969476375}{1001033261568} \) |
\( \bigl[i\) , \( 1\) , \( i\) , \( 16286\) , \( -1020323\bigr] \) |
${y}^2+i{x}{y}+i{y}={x}^{3}+{x}^{2}+16286{x}-1020323$ |
| 46818.2-d3 |
46818.2-d |
$4$ |
$6$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{36} \cdot 3^{16} \cdot 17^{6} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{4} \cdot 3^{4} \) |
$1$ |
$0.092415096$ |
3.326943479 |
\( \frac{46753267515625}{11591221248} \) |
\( \bigl[i\) , \( 1\) , \( i\) , \( -6754\) , \( -163235\bigr] \) |
${y}^2+i{x}{y}+i{y}={x}^{3}+{x}^{2}-6754{x}-163235$ |
| 46818.2-d4 |
46818.2-d |
$4$ |
$6$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{12} \cdot 3^{24} \cdot 17^{2} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$0.277245289$ |
3.326943479 |
\( \frac{1845026709625}{793152} \) |
\( \bigl[i\) , \( 1\) , \( i\) , \( -2299\) , \( 41857\bigr] \) |
${y}^2+i{x}{y}+i{y}={x}^{3}+{x}^{2}-2299{x}+41857$ |
| 46818.2-e1 |
46818.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2 \cdot 3^{14} \cdot 17^{7} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.169728025$ |
2.715648406 |
\( -\frac{1979660058649925}{501126} a - \frac{547309863864799}{167042} \) |
\( \bigl[1\) , \( -1\) , \( i\) , \( 14254 i + 723\) , \( 423947 i + 497800\bigr] \) |
${y}^2+{x}{y}+i{y}={x}^{3}-{x}^{2}+\left(14254i+723\right){x}+423947i+497800$ |
| 46818.2-e2 |
46818.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{2} \cdot 3^{16} \cdot 17^{8} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{5} \) |
$1$ |
$0.339456050$ |
2.715648406 |
\( \frac{9390341072075}{144825414} a - \frac{6456168132412}{217238121} \) |
\( \bigl[1\) , \( -1\) , \( i\) , \( 889 i + 48\) , \( 6527 i + 7750\bigr] \) |
${y}^2+{x}{y}+i{y}={x}^{3}-{x}^{2}+\left(889i+48\right){x}+6527i+7750$ |
| 46818.2-e3 |
46818.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2 \cdot 3^{14} \cdot 17^{13} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.169728025$ |
2.715648406 |
\( \frac{461275687224792005}{3495733423378566} a + \frac{140679328163848447}{1165244474459522} \) |
\( \bigl[i\) , \( 1\) , \( 1\) , \( 484 i - 627\) , \( -3179 i - 25732\bigr] \) |
${y}^2+i{x}{y}+{y}={x}^{3}+{x}^{2}+\left(484i-627\right){x}-3179i-25732$ |
| 46818.2-e4 |
46818.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{4} \cdot 3^{20} \cdot 17^{4} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.678912101$ |
2.715648406 |
\( -\frac{88739980}{132651} a + \frac{1762314767}{1591812} \) |
\( \bigl[i\) , \( 1\) , \( 1\) , \( 79 i + 48\) , \( -209 i + 188\bigr] \) |
${y}^2+i{x}{y}+{y}={x}^{3}+{x}^{2}+\left(79i+48\right){x}-209i+188$ |
| 46818.2-f1 |
46818.2-f |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{2} \cdot 3^{20} \cdot 17^{4} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.767433776$ |
6.139470211 |
\( \frac{46268279}{46818} \) |
\( \bigl[i\) , \( 1\) , \( i\) , \( 68\) , \( 201\bigr] \) |
${y}^2+i{x}{y}+i{y}={x}^{3}+{x}^{2}+68{x}+201$ |
| 46818.2-f2 |
46818.2-f |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
46818.2 |
\( 2 \cdot 3^{4} \cdot 17^{2} \) |
\( 2^{4} \cdot 3^{16} \cdot 17^{2} \) |
$2.62889$ |
$(a+1), (a+4), (a-4), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.534867552$ |
6.139470211 |
\( \frac{1771561}{612} \) |
\( \bigl[i\) , \( 1\) , \( i\) , \( -22\) , \( 21\bigr] \) |
${y}^2+i{x}{y}+i{y}={x}^{3}+{x}^{2}-22{x}+21$ |
*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.
