Elliptic curves over Q(−1)\Q(\sqrt{-1}) of conductor 31752.1

Results (26 matches)

  Download displayed columns for results to

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
31752.1-a1	31752.1-a	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{10} \cdot 3^{16} \cdot 7^{6}$	$2.38568$	$(a+1), (3), (7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3}$	$1$	$0.582969403$	1.165938807	$\frac{63821054}{3087} a - \frac{1625104}{343}$	$\bigl[i + 1$ , $i$ , $i + 1$ , $-19 i - 240$ , $98 i + 1438\bigr]$	${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-19i-240\right){x}+98i+1438$
31752.1-a2	31752.1-a	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{11} \cdot 3^{14} \cdot 7^{12}$	$2.38568$	$(a+1), (3), (7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$4$	$2^{2}$	$1$	$0.291484701$	1.165938807	$\frac{36618425}{352947} a - \frac{212113}{1029}$	$\bigl[i + 1$ , $i$ , $i + 1$ , $-289 i + 30$ , $-2224 i + 5164\bigr]$	${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-289i+30\right){x}-2224i+5164$
31752.1-b1	31752.1-b	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{4} \cdot 3^{6} \cdot 7^{4}$	$2.38568$	$(a+1), (3), (7)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{3}$	$0.154625521$	$3.339572940$	4.131065670	$\frac{11664}{49}$	$\bigl[i + 1$ , $i$ , $i + 1$ , $-i - 3$ , $-5 i\bigr]$	${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-i-3\right){x}-5i$
31752.1-b2	31752.1-b	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{8} \cdot 3^{6} \cdot 7^{2}$	$2.38568$	$(a+1), (3), (7)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{3}$	$0.154625521$	$3.339572940$	4.131065670	$\frac{55296}{7}$	$\bigl[0$ , $0$ , $0$ , $-6$ , $-5\bigr]$	${y}^2={x}^{3}-6{x}-5$
31752.1-c1	31752.1-c	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{8} \cdot 3^{14} \cdot 7^{8}$	$2.38568$	$(a+1), (3), (7)$	$2$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{5}$	$0.791183129$	$0.657452278$	4.161321206	$\frac{11696828}{7203}$	$\bigl[i + 1$ , $i$ , $0$ , $-108$ , $-162 i\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}-108{x}-162i$
31752.1-c2	31752.1-c	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{4} \cdot 3^{16} \cdot 7^{4}$	$2.38568$	$(a+1), (3), (7)$	$2$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	$2^{4}$	$0.791183129$	$1.314904556$	4.161321206	$\frac{810448}{441}$	$\bigl[i + 1$ , $i$ , $0$ , $27$ , $0\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+27{x}$
31752.1-c3	31752.1-c	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{8} \cdot 3^{14} \cdot 7^{2}$	$2.38568$	$(a+1), (3), (7)$	$2$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{4}$	$0.791183129$	$1.314904556$	4.161321206	$\frac{2725888}{21}$	$\bigl[0$ , $0$ , $0$ , $-66$ , $205\bigr]$	${y}^2={x}^{3}-66{x}+205$
31752.1-c4	31752.1-c	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{8} \cdot 3^{20} \cdot 7^{2}$	$2.38568$	$(a+1), (3), (7)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{3}$	$0.791183129$	$0.657452278$	4.161321206	$\frac{381775972}{567}$	$\bigl[i + 1$ , $i$ , $0$ , $342$ , $-2268 i\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+342{x}-2268i$
31752.1-d1	31752.1-d	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{8} \cdot 3^{6} \cdot 7^{4}$	$2.38568$	$(a+1), (3), (7)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{3}$	$0.257561760$	$2.574730348$	2.652608321	$-\frac{55296}{49}$	$\bigl[0$ , $0$ , $0$ , $-6$ , $9\bigr]$	${y}^2={x}^{3}-6{x}+9$
31752.1-d2	31752.1-d	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{4} \cdot 3^{6} \cdot 7^{2}$	$2.38568$	$(a+1), (3), (7)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{2}$	$0.515123520$	$2.574730348$	2.652608321	$\frac{21882096}{7}$	$\bigl[i + 1$ , $i$ , $0$ , $27$ , $70 i\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+27{x}+70i$
31752.1-e1	31752.1-e	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{4} \cdot 3^{12} \cdot 7^{2}$	$2.38568$	$(a+1), (3), (7)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{3}$	$0.284457192$	$2.677530478$	3.046571210	$\frac{432}{7}$	$\bigl[i + 1$ , $i$ , $i + 1$ , $-i - 3$ , $5 i\bigr]$	${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-i-3\right){x}+5i$
31752.1-e2	31752.1-e	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{10} \cdot 3^{12} \cdot 7^{8}$	$2.38568$	$(a+1), (3), (7)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{5}$	$0.284457192$	$0.669382619$	3.046571210	$\frac{11090466}{2401}$	$\bigl[i + 1$ , $i$ , $i + 1$ , $-i + 132$ , $-400 i\bigr]$	${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-i+132\right){x}-400i$
31752.1-e3	31752.1-e	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{8} \cdot 3^{12} \cdot 7^{4}$	$2.38568$	$(a+1), (3), (7)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	$2^{5}$	$0.568914384$	$1.338765239$	3.046571210	$\frac{740772}{49}$	$\bigl[i + 1$ , $i$ , $i + 1$ , $-i + 42$ , $122 i\bigr]$	${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-i+42\right){x}+122i$
31752.1-e4	31752.1-e	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{10} \cdot 3^{12} \cdot 7^{2}$	$2.38568$	$(a+1), (3), (7)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{3}$	$1.137828769$	$0.669382619$	3.046571210	$\frac{1443468546}{7}$	$\bigl[i + 1$ , $i$ , $i + 1$ , $-i + 672$ , $7052 i\bigr]$	${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-i+672\right){x}+7052i$
31752.1-f1	31752.1-f	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{8} \cdot 3^{18} \cdot 7^{8}$	$2.38568$	$(a+1), (3), (7)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{6}$	$1$	$0.463617288$	1.854469154	$-\frac{2725888}{64827}$	$\bigl[0$ , $0$ , $0$ , $-66$ , $-1339\bigr]$	${y}^2={x}^{3}-66{x}-1339$
31752.1-f2	31752.1-f	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{8} \cdot 3^{36} \cdot 7^{2}$	$2.38568$	$(a+1), (3), (7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$4$	$2^{3}$	$1$	$0.231808644$	1.854469154	$\frac{6522128932}{3720087}$	$\bigl[i + 1$ , $i$ , $0$ , $882$ , $1652 i\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+882{x}+1652i$
31752.1-f3	31752.1-f	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{4} \cdot 3^{24} \cdot 7^{4}$	$2.38568$	$(a+1), (3), (7)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$4$	$2^{4}$	$1$	$0.463617288$	1.854469154	$\frac{6940769488}{35721}$	$\bigl[i + 1$ , $i$ , $0$ , $567$ , $-4900 i\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+567{x}-4900i$
31752.1-f4	31752.1-f	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{8} \cdot 3^{18} \cdot 7^{2}$	$2.38568$	$(a+1), (3), (7)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$16$	$2^{3}$	$1$	$0.231808644$	1.854469154	$\frac{7080974546692}{189}$	$\bigl[i + 1$ , $i$ , $0$ , $9072$ , $-328090 i\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+9072{x}-328090i$
31752.1-g1	31752.1-g	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{10} \cdot 3^{16} \cdot 7^{6}$	$2.38568$	$(a+1), (3), (7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3}$	$1$	$0.582969403$	1.165938807	$-\frac{63821054}{3087} a - \frac{1625104}{343}$	$\bigl[i + 1$ , $i$ , $i + 1$ , $17 i - 240$ , $98 i - 1438\bigr]$	${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(17i-240\right){x}+98i-1438$
31752.1-g2	31752.1-g	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{11} \cdot 3^{14} \cdot 7^{12}$	$2.38568$	$(a+1), (3), (7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$4$	$2^{2}$	$1$	$0.291484701$	1.165938807	$-\frac{36618425}{352947} a - \frac{212113}{1029}$	$\bigl[i + 1$ , $i$ , $i + 1$ , $287 i + 30$ , $-2224 i - 5164\bigr]$	${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(287i+30\right){x}-2224i-5164$
31752.1-h1	31752.1-h	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{4} \cdot 3^{18} \cdot 7^{4}$	$2.38568$	$(a+1), (3), (7)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{3}$	$1.013053864$	$1.113190980$	4.510889699	$\frac{11664}{49}$	$\bigl[i + 1$ , $i$ , $0$ , $-21$ , $-98 i\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}-21{x}-98i$
31752.1-h2	31752.1-h	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{8} \cdot 3^{18} \cdot 7^{2}$	$2.38568$	$(a+1), (3), (7)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{2}$	$2.026107729$	$1.113190980$	4.510889699	$\frac{55296}{7}$	$\bigl[0$ , $0$ , $0$ , $-54$ , $-135\bigr]$	${y}^2={x}^{3}-54{x}-135$
31752.1-i1	31752.1-i	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{8} \cdot 3^{18} \cdot 7^{4}$	$2.38568$	$(a+1), (3), (7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{4}$	$1$	$0.858243449$	3.432973797	$-\frac{55296}{49}$	$\bigl[0$ , $0$ , $0$ , $-54$ , $243\bigr]$	${y}^2={x}^{3}-54{x}+243$
31752.1-i2	31752.1-i	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{4} \cdot 3^{18} \cdot 7^{2}$	$2.38568$	$(a+1), (3), (7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$4$	$2^{2}$	$1$	$0.858243449$	3.432973797	$\frac{21882096}{7}$	$\bigl[i + 1$ , $i$ , $i + 1$ , $-i + 249$ , $1643 i\bigr]$	${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-i+249\right){x}+1643i$
31752.1-j1	31752.1-j	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{8} \cdot 3^{12} \cdot 7^{2}$	$2.38568$	$(a+1), (3), (7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{3}$	$1$	$2.131913095$	4.263826191	$-\frac{4}{7}$	$\bigl[i + 1$ , $i$ , $0$ , $0$ , $14 i\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+14i$
31752.1-j2	31752.1-j	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	31752.1	$2^{3} \cdot 3^{4} \cdot 7^{2}$	$2^{10} \cdot 3^{12} \cdot 7^{4}$	$2.38568$	$(a+1), (3), (7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{4}$	$1$	$1.065956547$	4.263826191	$\frac{3543122}{49}$	$\bigl[i + 1$ , $i$ , $0$ , $90$ , $374 i\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+90{x}+374i$

  Download displayed columns for results to

  *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.