Elliptic curves over \(\Q(\zeta_{11})^+\) of conductor 121.1

Results (9 matches)

 

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
121.1-a1	121.1-a	$2$	$11$	\(\Q(\zeta_{11})^+\)	$5$	$[5, 0]$	121.1	\( 11^{2} \)	\( - 11^{4} \)	$17.46636$	$(a^2+a-2)$	0	$\Z/11\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$11$	11B.1.1[5]	$1$	\( 1 \)	$1$	$17090.60992$	1.16731165	\( -24729001 \)	\( \bigl[a^{4} - 3 a^{2} + 1\) , \( a^{4} - a^{3} - 3 a^{2} + 2 a\) , \( a^{2} + a - 1\) , \( -15 a^{4} + 38 a^{3} - 10 a^{2} - 19 a\) , \( 94 a^{4} - 262 a^{3} + 77 a^{2} + 174 a - 46\bigr] \)	${y}^2+\left(a^{4}-3a^{2}+1\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{4}-a^{3}-3a^{2}+2a\right){x}^{2}+\left(-15a^{4}+38a^{3}-10a^{2}-19a\right){x}+94a^{4}-262a^{3}+77a^{2}+174a-46$
121.1-a2	121.1-a	$2$	$11$	\(\Q(\zeta_{11})^+\)	$5$	$[5, 0]$	121.1	\( 11^{2} \)	\( - 11^{8} \)	$17.46636$	$(a^2+a-2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$11$	11B.1.10[5]	$1$	\( 1 \)	$1$	$141.2447101$	1.16731165	\( -121 \)	\( \bigl[a^{2} - 1\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( a^{4} + a^{3} - 4 a^{2} - 2 a + 3\) , \( 2 a^{4} - 5 a^{3} - 6 a^{2} + 13 a - 1\) , \( 16 a^{4} - 31 a^{3} - 40 a^{2} + 86 a - 22\bigr] \)	${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-2a+3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-3\right){x}^{2}+\left(2a^{4}-5a^{3}-6a^{2}+13a-1\right){x}+16a^{4}-31a^{3}-40a^{2}+86a-22$
121.1-b1	121.1-b	$2$	$11$	\(\Q(\zeta_{11})^+\)	$5$	$[5, 0]$	121.1	\( 11^{2} \)	\( - 11^{3} \)	$17.46636$	$(a^2+a-2)$	$1$	$\Z/11\Z$	$\textsf{potential}$	$-11$	$N(\mathrm{U}(1))$	✓	✓		✓	$11$	11B.1.1[5]	$1$	\( 2 \)	$0.089785156$	$28099.20145$	1.72316863	\( -32768 \)	\( \bigl[0\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 1\) , \( a^{3} + a^{2} - 2 a - 2\) , \( 9 a^{4} - 16 a^{3} - 23 a^{2} + 45 a - 10\) , \( -38 a^{4} + 67 a^{3} + 96 a^{2} - 188 a + 43\bigr] \)	${y}^2+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+3a+1\right){x}^{2}+\left(9a^{4}-16a^{3}-23a^{2}+45a-10\right){x}-38a^{4}+67a^{3}+96a^{2}-188a+43$
121.1-b2	121.1-b	$2$	$11$	\(\Q(\zeta_{11})^+\)	$5$	$[5, 0]$	121.1	\( 11^{2} \)	\( - 11^{9} \)	$17.46636$	$(a^2+a-2)$	$1$	$\mathsf{trivial}$	$\textsf{potential}$	$-11$	$N(\mathrm{U}(1))$	✓	✓		✓	$11$	11B.1.10[5]	$1$	\( 2 \)	$0.987636717$	$21.11134594$	1.72316863	\( -32768 \)	\( \bigl[0\) , \( -a^{4} + 4 a^{2} + a - 2\) , \( a^{4} - 4 a^{2} + 3\) , \( -4 a^{4} + 9 a^{3} - a^{2} - 3 a\) , \( -13 a^{4} + 34 a^{3} - 6 a^{2} - 25 a + 5\bigr] \)	${y}^2+\left(a^{4}-4a^{2}+3\right){y}={x}^{3}+\left(-a^{4}+4a^{2}+a-2\right){x}^{2}+\left(-4a^{4}+9a^{3}-a^{2}-3a\right){x}-13a^{4}+34a^{3}-6a^{2}-25a+5$
121.1-c1	121.1-c	$3$	$25$	\(\Q(\zeta_{11})^+\)	$5$	$[5, 0]$	121.1	\( 11^{2} \)	\( - 11^{11} \)	$17.46636$	$(a^2+a-2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$5$	5B.4.2	$1$	\( 2 \)	$1$	$63.74661725$	1.05366309	\( -\frac{52893159101157376}{11} \)	\( \bigl[0\) , \( a - 1\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -7820 a^{2} - 31282 a - 31281\) , \( -a^{4} + 263579 a^{3} + 1589301 a^{2} + 3194239 a + 2139918\bigr] \)	${y}^2+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-7820a^{2}-31282a-31281\right){x}-a^{4}+263579a^{3}+1589301a^{2}+3194239a+2139918$
121.1-c2	121.1-c	$3$	$25$	\(\Q(\zeta_{11})^+\)	$5$	$[5, 0]$	121.1	\( 11^{2} \)	\( - 11^{31} \)	$17.46636$	$(a^2+a-2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$5$	5Cs.4.1	$1$	\( 2 \)	$1$	$63.74661725$	1.05366309	\( -\frac{122023936}{161051} \)	\( \bigl[0\) , \( a - 1\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -10 a^{2} - 42 a - 41\) , \( -a^{4} + 19 a^{3} + 131 a^{2} + 279 a + 198\bigr] \)	${y}^2+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-10a^{2}-42a-41\right){x}-a^{4}+19a^{3}+131a^{2}+279a+198$
121.1-c3	121.1-c	$3$	$25$	\(\Q(\zeta_{11})^+\)	$5$	$[5, 0]$	121.1	\( 11^{2} \)	\( - 11^{11} \)	$17.46636$	$(a^2+a-2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$5$	5B.4.1	$1$	\( 2 \)	$1$	$63.74661725$	1.05366309	\( -\frac{4096}{11} \)	\( \bigl[0\) , \( a - 1\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -2 a - 1\) , \( -a^{4} - a^{3} + a^{2} - a - 2\bigr] \)	${y}^2+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2a-1\right){x}-a^{4}-a^{3}+a^{2}-a-2$
121.1-d1	121.1-d	$2$	$11$	\(\Q(\zeta_{11})^+\)	$5$	$[5, 0]$	121.1	\( 11^{2} \)	\( - 11^{10} \)	$17.46636$	$(a^2+a-2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$11$	11B.1.10[5]	$121$	\( 1 \)	$1$	$1.102971008$	1.10297101	\( -24729001 \)	\( \bigl[1\) , \( 1\) , \( 1\) , \( -30\) , \( -76\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-30{x}-76$
121.1-d2	121.1-d	$2$	$11$	\(\Q(\zeta_{11})^+\)	$5$	$[5, 0]$	121.1	\( 11^{2} \)	\( - 11^{2} \)	$17.46636$	$(a^2+a-2)$	0	$\Z/11\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$11$	11B.1.1[5]	$1$	\( 1 \)	$1$	$16148.59853$	1.10297101	\( -121 \)	\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{4} - a^{3} + 5 a^{2} + 4 a - 5\) , \( a^{4} - 3 a^{2}\) , \( -3 a^{4} + 2 a^{3} + 9 a^{2} - 5 a\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 1\bigr] \)	${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{4}-3a^{2}\right){y}={x}^{3}+\left(-a^{4}-a^{3}+5a^{2}+4a-5\right){x}^{2}+\left(-3a^{4}+2a^{3}+9a^{2}-5a\right){x}+a^{4}-2a^{3}-3a^{2}+6a+1$

 

  *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.