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
4.1-a3
4.1-a
$6$
$45$
4.4.10512.1
$4$
$[4, 0]$
4.1
\( 2^{2} \)
\( 2^{30} \)
$10.89529$
$(a^3-a^2-5a)$
$1$
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3, 5$
3Cs.1.1 , 5B.4.2
$1$
\( 3 \cdot 5 \)
$1.075821811$
$28.65947154$
2.004816315
\( -\frac{266320696004661}{256} a^{3} + \frac{266320696004661}{256} a^{2} + \frac{1331603480023305}{256} a + \frac{90950209108247}{32} \)
\( \bigl[a^{2} - a - 4\) , \( a\) , \( a^{3} - 5 a - 5\) , \( 29 a^{3} - 28 a^{2} - 147 a - 51\) , \( 12 a^{3} + 63 a^{2} - 206 a - 287\bigr] \)
${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{3}-5a-5\right){y}={x}^{3}+a{x}^{2}+\left(29a^{3}-28a^{2}-147a-51\right){x}+12a^{3}+63a^{2}-206a-287$
4.1-b3
4.1-b
$6$
$45$
4.4.10512.1
$4$
$[4, 0]$
4.1
\( 2^{2} \)
\( 2^{30} \)
$10.89529$
$(a^3-a^2-5a)$
$1$
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3, 5$
3Cs.1.1 , 5B.4.2
$1$
\( 3 \cdot 5 \)
$1.075821811$
$28.65947154$
2.004816315
\( -\frac{266320696004661}{256} a^{3} + \frac{266320696004661}{256} a^{2} + \frac{1331603480023305}{256} a + \frac{90950209108247}{32} \)
\( \bigl[a^{3} - 6 a - 4\) , \( a^{3} - 6 a - 6\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 29 a^{3} - 27 a^{2} - 144 a - 50\) , \( -11 a^{3} - 34 a^{2} + 184 a + 207\bigr] \)
${y}^2+\left(a^{3}-6a-4\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(a^{3}-6a-6\right){x}^{2}+\left(29a^{3}-27a^{2}-144a-50\right){x}-11a^{3}-34a^{2}+184a+207$
4.1-c4
4.1-c
$6$
$45$
4.4.10512.1
$4$
$[4, 0]$
4.1
\( 2^{2} \)
\( 2^{30} \)
$10.89529$
$(a^3-a^2-5a)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3, 5$
3Cs , 5B.4.2
$1$
\( 1 \)
$1$
$157.8350663$
1.539433102
\( -\frac{266320696004661}{256} a^{3} + \frac{266320696004661}{256} a^{2} + \frac{1331603480023305}{256} a + \frac{90950209108247}{32} \)
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( a - 1\) , \( 1\) , \( 1949 a^{3} - 2435 a^{2} - 10586 a + 1554\) , \( 48278 a^{3} - 60389 a^{2} - 262368 a + 38590\bigr] \)
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(1949a^{3}-2435a^{2}-10586a+1554\right){x}+48278a^{3}-60389a^{2}-262368a+38590$
4.1-d1
4.1-d
$6$
$45$
4.4.10512.1
$4$
$[4, 0]$
4.1
\( 2^{2} \)
\( 2^{30} \)
$10.89529$
$(a^3-a^2-5a)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3, 5$
3Cs , 5B.1.2
$25$
\( 1 \)
$1$
$5.203946933$
1.268908164
\( -\frac{266320696004661}{256} a^{3} + \frac{266320696004661}{256} a^{2} + \frac{1331603480023305}{256} a + \frac{90950209108247}{32} \)
\( \bigl[a^{3} - 5 a - 4\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 1947 a^{3} - 2429 a^{2} - 10567 a + 1563\) , \( -49250 a^{3} + 61634 a^{2} + 267701 a - 39371\bigr] \)
${y}^2+\left(a^{3}-5a-4\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+2\right){x}^{2}+\left(1947a^{3}-2429a^{2}-10567a+1563\right){x}-49250a^{3}+61634a^{2}+267701a-39371$
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*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.