Elliptic curves over \(\Q(\sqrt{106}) \)

Results (12 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
1.1-a1	1.1-a	$1$	$1$	\(\Q(\sqrt{106}) \)	$2$	$[2, 0]$	1.1	\( 1 \)	\( -1 \)	$1.84002$	$\textsf{none}$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓	✓			$1$	\( 1 \)	$1$	$33.67140610$	1.635228035	\( 25777 a + 264235 \)	\( \bigl[1\) , \( a\) , \( 1\) , \( -262494 a - 2702508\) , \( -235431668 a - 2423917415\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-262494a-2702508\right){x}-235431668a-2423917415$
1.1-b1	1.1-b	$1$	$1$	\(\Q(\sqrt{106}) \)	$2$	$[2, 0]$	1.1	\( 1 \)	\( -1 \)	$1.84002$	$\textsf{none}$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓	✓			$1$	\( 1 \)	$1$	$33.67140610$	1.635228035	\( -25777 a + 264235 \)	\( \bigl[1\) , \( -a\) , \( 1\) , \( 262494 a - 2702508\) , \( 235431668 a - 2423917415\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(262494a-2702508\right){x}+235431668a-2423917415$
1.1-c1	1.1-c	$1$	$1$	\(\Q(\sqrt{106}) \)	$2$	$[2, 0]$	1.1	\( 1 \)	\( - 2^{12} \)	$1.84002$	$\textsf{none}$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓	✓			$1$	\( 1 \)	$1$	$33.67140610$	1.635228035	\( -25777 a + 264235 \)	\( \bigl[a\) , \( a + 1\) , \( a\) , \( 1049995 a - 10809937\) , \( 1882093082 a - 19377332290\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(1049995a-10809937\right){x}+1882093082a-19377332290$
1.1-d1	1.1-d	$1$	$1$	\(\Q(\sqrt{106}) \)	$2$	$[2, 0]$	1.1	\( 1 \)	\( - 2^{12} \)	$1.84002$	$\textsf{none}$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓	✓			$1$	\( 1 \)	$1$	$33.67140610$	1.635228035	\( 25777 a + 264235 \)	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -1049995 a - 10809937\) , \( -1882093082 a - 19377332290\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1049995a-10809937\right){x}-1882093082a-19377332290$
3.1-a1	3.1-a	$1$	$1$	\(\Q(\sqrt{106}) \)	$2$	$[2, 0]$	3.1	\( 3 \)	\( 3^{12} \)	$2.42160$	$(3,a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 2 \)	$1$	$7.279780841$	0.707074821	\( \frac{917674496}{531441} a - \frac{12306610112}{531441} \)	\( \bigl[a\) , \( -a\) , \( 1\) , \( -301050 a - 3099043\) , \( 329963957 a + 3397188706\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-301050a-3099043\right){x}+329963957a+3397188706$
3.1-b1	3.1-b	$1$	$1$	\(\Q(\sqrt{106}) \)	$2$	$[2, 0]$	3.1	\( 3 \)	\( 2^{12} \cdot 3^{4} \)	$2.42160$	$(3,a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 2^{2} \)	$1.056683201$	$12.28540203$	5.043606958	\( \frac{78884}{81} a - \frac{804137}{81} \)	\( \bigl[a\) , \( a + 1\) , \( a\) , \( 19 a + 241\) , \( 100 a + 1190\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(19a+241\right){x}+100a+1190$
3.1-c1	3.1-c	$1$	$1$	\(\Q(\sqrt{106}) \)	$2$	$[2, 0]$	3.1	\( 3 \)	\( 3^{4} \)	$2.42160$	$(3,a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 2 \)	$1$	$12.28540203$	1.193263731	\( \frac{78884}{81} a - \frac{804137}{81} \)	\( \bigl[1\) , \( -a\) , \( 0\) , \( 37\) , \( -4 a + 2\bigr] \)	${y}^2+{x}{y}={x}^{3}-a{x}^{2}+37{x}-4a+2$
3.1-d1	3.1-d	$1$	$1$	\(\Q(\sqrt{106}) \)	$2$	$[2, 0]$	3.1	\( 3 \)	\( 2^{12} \cdot 3^{12} \)	$2.42160$	$(3,a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 2^{2} \cdot 3 \)	$0.879054255$	$7.279780841$	7.458685570	\( \frac{917674496}{531441} a - \frac{12306610112}{531441} \)	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 2738701812 a - 28196660894\) , \( 255545025319958 a - 2630997065063416\bigr] \)	${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(2738701812a-28196660894\right){x}+255545025319958a-2630997065063416$
3.2-a1	3.2-a	$1$	$1$	\(\Q(\sqrt{106}) \)	$2$	$[2, 0]$	3.2	\( 3 \)	\( 3^{12} \)	$2.42160$	$(3,a+2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 2 \)	$1$	$7.279780841$	0.707074821	\( -\frac{917674496}{531441} a - \frac{12306610112}{531441} \)	\( \bigl[a\) , \( a\) , \( 1\) , \( 301049 a - 3099043\) , \( -329963957 a + 3397188706\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(301049a-3099043\right){x}-329963957a+3397188706$
3.2-b1	3.2-b	$1$	$1$	\(\Q(\sqrt{106}) \)	$2$	$[2, 0]$	3.2	\( 3 \)	\( 2^{12} \cdot 3^{4} \)	$2.42160$	$(3,a+2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 2^{2} \)	$1.056683201$	$12.28540203$	5.043606958	\( -\frac{78884}{81} a - \frac{804137}{81} \)	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -19 a + 241\) , \( -100 a + 1190\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-19a+241\right){x}-100a+1190$
3.2-c1	3.2-c	$1$	$1$	\(\Q(\sqrt{106}) \)	$2$	$[2, 0]$	3.2	\( 3 \)	\( 3^{4} \)	$2.42160$	$(3,a+2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 2 \)	$1$	$12.28540203$	1.193263731	\( -\frac{78884}{81} a - \frac{804137}{81} \)	\( \bigl[1\) , \( a\) , \( 0\) , \( 37\) , \( 4 a + 2\bigr] \)	${y}^2+{x}{y}={x}^{3}+a{x}^{2}+37{x}+4a+2$
3.2-d1	3.2-d	$1$	$1$	\(\Q(\sqrt{106}) \)	$2$	$[2, 0]$	3.2	\( 3 \)	\( 2^{12} \cdot 3^{12} \)	$2.42160$	$(3,a+2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 2^{2} \cdot 3 \)	$0.879054255$	$7.279780841$	7.458685570	\( -\frac{917674496}{531441} a - \frac{12306610112}{531441} \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -2738701812 a - 28196660894\) , \( -255545025319958 a - 2630997065063416\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2738701812a-28196660894\right){x}-255545025319958a-2630997065063416$

 

  *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.