Elliptic curve search results

Results (14 matches)

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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
6.4-a4	6.4-a	$8$	$28$	\(\Q(\sqrt{-23}) \)	$2$	$[0, 1]$	6.4	\( 2 \cdot 3 \)	\( 2^{28} \cdot 3^{7} \)	$0.67072$	$(2,a+1), (3,a+2)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 7$	2B, 7B.2.3	$1$	\( 2^{2} \cdot 7 \)	$1$	$1.424700212$	1.039746854	\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \)	\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -4 a - 17\) , \( -9 a\bigr] \)	${y}^2+{x}{y}={x}^3+\left(-a-1\right){x}^2+\left(-4a-17\right){x}-9a$
144.15-a4	144.15-a	$8$	$28$	\(\Q(\sqrt{-23}) \)	$2$	$[0, 1]$	144.15	\( 2^{4} \cdot 3^{2} \)	\( 2^{52} \cdot 3^{13} \)	$1.48454$	$(2,a+1), (3,a+2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 7$	2B, 7B	$1$	\( 2^{3} \)	$2.149866424$	$0.411275525$	1.474926613	\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \)	\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 50 a - 1027\) , \( -2635 a - 10438\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(50a-1027\right){x}-2635a-10438$
162.8-a4	162.8-a	$8$	$28$	\(\Q(\sqrt{-23}) \)	$2$	$[0, 1]$	162.8	\( 2 \cdot 3^{4} \)	\( 2^{28} \cdot 3^{19} \)	$1.52891$	$(2,a+1), (3,a), (3,a+2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 7$	2B, 7B	$1$	\( 2^{4} \)	$0.749127778$	$0.474900070$	1.186900192	\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \)	\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -44 a - 137\) , \( 467 a - 104\bigr] \)	${y}^2+{x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(-44a-137\right){x}+467a-104$
192.14-a4	192.14-a	$8$	$28$	\(\Q(\sqrt{-23}) \)	$2$	$[0, 1]$	192.14	\( 2^{6} \cdot 3 \)	\( 2^{46} \cdot 3^{7} \)	$1.59525$	$(2,a+1), (3,a+2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 7$	2B, 7B	$1$	\( 2^{2} \)	$3.088007485$	$0.503707590$	1.297337325	\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \)	\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -37 a + 167\) , \( -362 a - 122\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-37a+167\right){x}-362a-122$
288.6-c4	288.6-c	$8$	$28$	\(\Q(\sqrt{-23}) \)	$2$	$[0, 1]$	288.6	\( 2^{5} \cdot 3^{2} \)	\( 2^{40} \cdot 3^{13} \)	$1.76543$	$(2,a), (2,a+1), (3,a+2)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 7$	2B, 7B	$1$	\( 2^{5} \cdot 7 \)	$1$	$0.411275525$	2.401192506	\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \)	\( \bigl[a\) , \( -a\) , \( a\) , \( -90 a + 178\) , \( 446 a + 923\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3-a{x}^2+\left(-90a+178\right){x}+446a+923$
384.4-b4	384.4-b	$8$	$28$	\(\Q(\sqrt{-23}) \)	$2$	$[0, 1]$	384.4	\( 2^{7} \cdot 3 \)	\( 2^{46} \cdot 3^{7} \)	$1.89708$	$(2,a), (2,a+1), (3,a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 7$	2B, 7B	$1$	\( 2^{2} \cdot 7 \)	$1$	$0.503707590$	1.470424103	\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \)	\( \bigl[a\) , \( 0\) , \( a\) , \( 69 a - 53\) , \( -418 a + 210\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+\left(69a-53\right){x}-418a+210$
432.18-c4	432.18-c	$8$	$28$	\(\Q(\sqrt{-23}) \)	$2$	$[0, 1]$	432.18	\( 2^{4} \cdot 3^{3} \)	\( 2^{40} \cdot 3^{13} \)	$1.95377$	$(2,a+1), (3,a), (3,a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 7$	2B, 7B	$1$	\( 2^{3} \cdot 7 \)	$1$	$0.411275525$	2.401192506	\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \)	\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 84 a - 195\) , \( 382 a - 1582\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(84a-195\right){x}+382a-1582$
576.21-c4	576.21-c	$8$	$28$	\(\Q(\sqrt{-23}) \)	$2$	$[0, 1]$	576.21	\( 2^{6} \cdot 3^{2} \)	\( 2^{58} \cdot 3^{13} \)	$2.09946$	$(2,a+1), (3,a+2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 7$	2B, 7B	$1$	\( 2^{3} \)	$7.896528301$	$0.290815713$	3.830717567	\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 707 a + 751\) , \( -8150 a - 28854\bigr] \)	${y}^2={x}^3+\left(707a+751\right){x}-8150a-28854$
768.10-b4	768.10-b	$8$	$28$	\(\Q(\sqrt{-23}) \)	$2$	$[0, 1]$	768.10	\( 2^{8} \cdot 3 \)	\( 2^{52} \cdot 3^{7} \)	$2.25602$	$(2,a), (2,a+1), (3,a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 7$	2B, 7B	$1$	\( 2^{3} \cdot 7 \)	$1$	$0.356175053$	2.079493709	\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \)	\( \bigl[0\) , \( a\) , \( 0\) , \( -77 a - 245\) , \( 976 a - 176\bigr] \)	${y}^2={x}^3+a{x}^2+\left(-77a-245\right){x}+976a-176$
864.6-a4	864.6-a	$8$	$28$	\(\Q(\sqrt{-23}) \)	$2$	$[0, 1]$	864.6	\( 2^{5} \cdot 3^{3} \)	\( 2^{52} \cdot 3^{13} \)	$2.32344$	$(2,a), (2,a+1), (3,a), (3,a+2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 7$	2B, 7B	$1$	\( 2^{3} \)	$3.340771694$	$0.411275525$	2.291953130	\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \)	\( \bigl[a\) , \( a\) , \( 0\) , \( -406 a - 81\) , \( -3148 a + 13865\bigr] \)	${y}^2+a{x}{y}={x}^3+a{x}^2+\left(-406a-81\right){x}-3148a+13865$
1014.10-a4	1014.10-a	$8$	$28$	\(\Q(\sqrt{-23}) \)	$2$	$[0, 1]$	1014.10	\( 2 \cdot 3 \cdot 13^{2} \)	\( 2^{40} \cdot 3^{7} \cdot 13^{6} \)	$2.41831$	$(2,a+1), (3,a+2), (13,a+4)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 7$	2B, 7B	$1$	\( 2^{3} \cdot 7^{2} \)	$0.129566197$	$0.395140743$	4.184712962	\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \)	\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 193 a + 890\) , \( 2302 a - 16890\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(193a+890\right){x}+2302a-16890$
1014.12-c4	1014.12-c	$8$	$28$	\(\Q(\sqrt{-23}) \)	$2$	$[0, 1]$	1014.12	\( 2 \cdot 3 \cdot 13^{2} \)	\( 2^{40} \cdot 3^{7} \cdot 13^{6} \)	$2.41831$	$(2,a+1), (3,a+2), (13,a+8)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 7$	2B, 7B	$1$	\( 2^{2} \cdot 7 \)	$1$	$0.395140743$	1.153495568	\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \)	\( \bigl[a\) , \( 0\) , \( a\) , \( 301 a + 676\) , \( 737 a + 14253\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+\left(301a+676\right){x}+737a+14253$
1152.6-b4	1152.6-b	$8$	$28$	\(\Q(\sqrt{-23}) \)	$2$	$[0, 1]$	1152.6	\( 2^{7} \cdot 3^{2} \)	\( 2^{58} \cdot 3^{13} \)	$2.49670$	$(2,a), (2,a+1), (3,a+2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 7$	2B, 7B	$1$	\( 2^{4} \)	$2.603516192$	$0.290815713$	2.526005057	\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \)	\( \bigl[a\) , \( -a\) , \( 0\) , \( -73 a - 1979\) , \( 11411 a + 23519\bigr] \)	${y}^2+a{x}{y}={x}^3-a{x}^2+\left(-73a-1979\right){x}+11411a+23519$
1728.26-d4	1728.26-d	$8$	$28$	\(\Q(\sqrt{-23}) \)	$2$	$[0, 1]$	1728.26	\( 2^{6} \cdot 3^{3} \)	\( 2^{58} \cdot 3^{13} \)	$2.76305$	$(2,a+1), (3,a), (3,a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 7$	2B, 7B	$1$	\( 2^{3} \cdot 7 \)	$1$	$0.290815713$	1.697899503	\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \)	\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -746 a - 590\) , \( 12388 a - 26756\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(-746a-590\right){x}+12388a-26756$

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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.