Elliptic curves over 3.3.148.1 of conductor 475.2

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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
475.2-a1	475.2-a	$4$	$10$	3.3.148.1	$3$	$[3, 0]$	475.2	\( 5^{2} \cdot 19 \)	\( - 5^{6} \cdot 19^{2} \)	$3.03658$	$(a^2-a-1), (-a^2-a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 5$	2B, 5B.4.1	$1$	\( 2^{2} \)	$1$	$39.08186094$	3.212509180	\( \frac{19422176}{361} a^{2} - \frac{29688224}{361} a + \frac{7279536}{361} \)	\( \bigl[a^{2} - a - 2\) , \( a^{2} - 2 a - 2\) , \( 1\) , \( -3 a^{2} + 5 a\) , \( -7 a^{2} + 20 a - 6\bigr] \)	${y}^2+\left(a^{2}-a-2\right){x}{y}+{y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(-3a^{2}+5a\right){x}-7a^{2}+20a-6$
475.2-a2	475.2-a	$4$	$10$	3.3.148.1	$3$	$[3, 0]$	475.2	\( 5^{2} \cdot 19 \)	\( - 5^{6} \cdot 19^{10} \)	$3.03658$	$(a^2-a-1), (-a^2-a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 5$	2B, 5B.4.2	$1$	\( 2^{2} \cdot 5 \)	$1$	$7.816372189$	3.212509180	\( \frac{34600248516351310880}{6131066257801} a^{2} + \frac{14707001073639931296}{6131066257801} a - \frac{7931450838072839760}{6131066257801} \)	\( \bigl[a^{2} - a - 2\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -841117 a^{2} - 984167 a + 387592\) , \( -773119562 a^{2} - 904616786 a + 356262107\bigr] \)	${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(-841117a^{2}-984167a+387592\right){x}-773119562a^{2}-904616786a+356262107$
475.2-a3	475.2-a	$4$	$10$	3.3.148.1	$3$	$[3, 0]$	475.2	\( 5^{2} \cdot 19 \)	\( - 5^{6} \cdot 19 \)	$3.03658$	$(a^2-a-1), (-a^2-a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 5$	2B, 5B.4.1	$1$	\( 2 \)	$1$	$78.16372189$	3.212509180	\( -\frac{2985984}{19} a^{2} + \frac{2048000}{19} a + \frac{9617408}{19} \)	\( \bigl[0\) , \( a^{2} - 2 a - 3\) , \( a^{2} - 2\) , \( -27039517 a^{2} - 31638572 a + 12460114\) , \( 2938778407350 a^{2} + 3438624901627 a - 1354221790452\bigr] \)	${y}^2+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(-27039517a^{2}-31638572a+12460114\right){x}+2938778407350a^{2}+3438624901627a-1354221790452$
475.2-a4	475.2-a	$4$	$10$	3.3.148.1	$3$	$[3, 0]$	475.2	\( 5^{2} \cdot 19 \)	\( - 5^{6} \cdot 19^{5} \)	$3.03658$	$(a^2-a-1), (-a^2-a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 5$	2B, 5B.4.2	$1$	\( 2 \cdot 5 \)	$1$	$15.63274437$	3.212509180	\( \frac{21475865428001583104}{2476099} a^{2} - \frac{53285799291084800000}{2476099} a + \frac{14499029076062806016}{2476099} \)	\( \bigl[0\) , \( a - 1\) , \( a^{2} - 2\) , \( -160804400212360 a^{2} - 188155055678488 a + 74100456918827\) , \( -2383827087438207587333 a^{2} - 2789283861464543203831 a + 1098494047186335627733\bigr] \)	${y}^2+\left(a^{2}-2\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-160804400212360a^{2}-188155055678488a+74100456918827\right){x}-2383827087438207587333a^{2}-2789283861464543203831a+1098494047186335627733$
475.2-b1	475.2-b	$4$	$6$	3.3.148.1	$3$	$[3, 0]$	475.2	\( 5^{2} \cdot 19 \)	\( - 5^{6} \cdot 19^{2} \)	$3.03658$	$(a^2-a-1), (-a^2-a-1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2^{2} \)	$0.068895405$	$125.0048800$	2.123770703	\( \frac{309847904}{361} a^{2} + \frac{362105728}{361} a - \frac{142642480}{361} \)	\( \bigl[a^{2} - a - 2\) , \( -a - 1\) , \( a^{2} - a - 1\) , \( -114167 a^{2} - 133585 a + 52608\) , \( 38795288 a^{2} + 45393843 a - 17877301\bigr] \)	${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-114167a^{2}-133585a+52608\right){x}+38795288a^{2}+45393843a-17877301$
475.2-b2	475.2-b	$4$	$6$	3.3.148.1	$3$	$[3, 0]$	475.2	\( 5^{2} \cdot 19 \)	\( - 5^{6} \cdot 19^{6} \)	$3.03658$	$(a^2-a-1), (-a^2-a-1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2^{2} \)	$0.206686216$	$41.66829336$	2.123770703	\( \frac{15636831814539296}{47045881} a^{2} - \frac{38648046941482112}{47045881} a + \frac{10231591814174352}{47045881} \)	\( \bigl[a^{2} - a - 2\) , \( -a - 1\) , \( a^{2} - a - 1\) , \( -274212 a^{2} - 320850 a + 126358\) , \( -90112799 a^{2} - 105439768 a + 41524980\bigr] \)	${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-274212a^{2}-320850a+126358\right){x}-90112799a^{2}-105439768a+41524980$
475.2-b3	475.2-b	$4$	$6$	3.3.148.1	$3$	$[3, 0]$	475.2	\( 5^{2} \cdot 19 \)	\( - 5^{6} \cdot 19^{3} \)	$3.03658$	$(a^2-a-1), (-a^2-a-1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2 \)	$0.413372433$	$41.66829336$	2.123770703	\( -\frac{72241514230136832}{6859} a^{2} + \frac{49766614320668672}{6859} a + \frac{232207325676879872}{6859} \)	\( \bigl[0\) , \( -a^{2} + a + 3\) , \( a\) , \( -25 a^{2} + 79 a - 29\) , \( 165 a^{2} - 431 a + 121\bigr] \)	${y}^2+a{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-25a^{2}+79a-29\right){x}+165a^{2}-431a+121$
475.2-b4	475.2-b	$4$	$6$	3.3.148.1	$3$	$[3, 0]$	475.2	\( 5^{2} \cdot 19 \)	\( - 5^{6} \cdot 19 \)	$3.03658$	$(a^2-a-1), (-a^2-a-1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2 \)	$0.137790811$	$125.0048800$	2.123770703	\( -\frac{16384}{19} a^{2} - \frac{319488}{19} a + \frac{737280}{19} \)	\( \bigl[0\) , \( -a^{2} + a + 3\) , \( a\) , \( -5 a^{2} + 9 a + 1\) , \( -6 a^{2} + 15 a - 3\bigr] \)	${y}^2+a{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-5a^{2}+9a+1\right){x}-6a^{2}+15a-3$

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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.