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{
"0": {
"comment": "The continuous modified period is exactly $\\widetilde N(0)=0$, unlike the ordinary value HREF{T649#0}[$N(0)$]. This is an intrinsic specialization, not a nonsingular CM case of the model $C_t$."
},
"24": {
"comment": "The rational CM parameter has order discriminant $-27$; its modified period is negative. At $t=t_Q([9,3,1])$, $C_t$ has CM order discriminant $\\Delta=-27$. The parameter is rational. CITE{ST}, Section 1. This case also belongs to CITE{S}, Table 2; its modified-period identity is proved in CITE{TG}, Theorem 1.5."
},
"18-6*sqrt(3)": {
"param-latex": "$18-6\\sqrt{3}$",
"comment": "At $t=t_Q([9,6,2])$, $C_t$ has CM order discriminant $\\Delta=-36$. The parameter is real quadratic. CITE{ST}, Table 1, row 14."
},
"-27/2+27/2*sqrt(5)": {
"param-latex": "$\\frac{-27}{2}+\\frac{27}{2}\\sqrt{5}$",
"comment": "At $t=t_Q([6,3,1])$, $C_t$ has CM order discriminant $\\Delta=-15$. The parameter is real quadratic. CITE{ST}, Table 1, row 17."
},
"-4320+1944*sqrt(5)": {
"param-latex": "$-4320+1944\\sqrt{5}$",
"comment": "At $t=t_Q([21,3,1])$, $C_t$ has CM order discriminant $\\Delta=-75$. The parameter is real quadratic. CITE{ST}, Table 1, row 19."
},
"-864+216*sqrt(17)": {
"param-latex": "$-864+216\\sqrt{17}$",
"comment": "At $t=t_Q([15,3,1])$, $C_t$ has CM order discriminant $\\Delta=-51$. The parameter is real quadratic. CITE{ST}, Table 1, row 22."
},
"-163296+35640*sqrt(21)": {
"param-latex": "$-163296+35640\\sqrt{21}$",
"comment": "At $t=t_Q([39,3,1])$, $C_t$ has CM order discriminant $\\Delta=-147$. The parameter is real quadratic. CITE{ST}, Table 1, row 23."
},
"4+4*sqrt(33)": {
"param-latex": "$4+4\\sqrt{33}$",
"comment": "At $t=t_Q([27,3,1])$, $C_t$ has CM order discriminant $\\Delta=-99$. The parameter is real quadratic. CITE{ST}, Table 1, row 24."
},
"-16740+2916*sqrt(33)": {
"param-latex": "$-16740+2916\\sqrt{33}$",
"comment": "At $t=t_Q([5,3,1])$, $C_t$ has CM order discriminant $\\Delta=-11$. The parameter is real quadratic. CITE{ST}, Table 1, row 25."
},
"-55296+8640*sqrt(41)": {
"param-latex": "$-55296+8640\\sqrt{41}$",
"comment": "At $t=t_Q([33,3,1])$, $C_t$ has CM order discriminant $\\Delta=-123$. The parameter is real quadratic. CITE{ST}, Table 1, row 26."
},
"-13500000+1431000*sqrt(89)": {
"param-latex": "$-13500000+1431000\\sqrt{89}$",
"comment": "At $t=t_Q([69,3,1])$, $C_t$ has CM order discriminant $\\Delta=-267$. The parameter is real quadratic. CITE{ST}, Table 1, row 27."
},
"1": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"2": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"3": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"4": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"5": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"6": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"7": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"8": {
"comment": "Here $k=2$; the modified-period identity with the conductor-19 elliptic curve is proved in CITE{S}, Theorem 2."
},
"9": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"10": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"11": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"12": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"13": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"14": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"15": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"16": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"17": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"18": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"19": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"20": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"21": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"22": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"23": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"25": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"26": {
"comment": "An integer parameter of the modified-period arithmetic family in CITE{S}, Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table."
},
"cm-m243-63-3-1": {
"param-latex": "$t_Q([63,3,1])$",
"comment": "At $t=t_Q([63,3,1])$, $C_t$ has CM order discriminant $\\Delta=-243$. The parameter is real cubic; its other two algebraic conjugates are nonreal. CITE{Supplement}, entry ν-C-3-1."
},
"cm-m288-27-24-8": {
"param-latex": "$t_Q([27,24,8])$",
"comment": "At $t=t_Q([27,24,8])$, $C_t$ has CM order discriminant $\\Delta=-288$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 52."
},
"cm-m288-27-12-4": {
"param-latex": "$t_Q([27,12,4])$",
"comment": "At $t=t_Q([27,12,4])$, $C_t$ has CM order discriminant $\\Delta=-288$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 52."
},
"cm-m96-15-12-4": {
"param-latex": "$t_Q([15,12,4])$",
"comment": "At $t=t_Q([15,12,4])$, $C_t$ has CM order discriminant $\\Delta=-96$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 53."
},
"cm-m192-21-12-4": {
"param-latex": "$t_Q([21,12,4])$",
"comment": "At $t=t_Q([21,12,4])$, $C_t$ has CM order discriminant $\\Delta=-192$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 54."
},
"cm-m32-8-8-3": {
"param-latex": "$t_Q([8,8,3])$",
"comment": "At $t=t_Q([8,8,3])$, $C_t$ has CM order discriminant $\\Delta=-32$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 55."
},
"cm-m180-18-18-7": {
"param-latex": "$t_Q([18,18,7])$",
"comment": "At $t=t_Q([18,18,7])$, $C_t$ has CM order discriminant $\\Delta=-180$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 57."
},
"cm-m180-27-6-2": {
"param-latex": "$t_Q([27,6,2])$",
"comment": "At $t=t_Q([27,6,2])$, $C_t$ has CM order discriminant $\\Delta=-180$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 57."
},
"cm-m20-7-6-2": {
"param-latex": "$t_Q([7,6,2])$",
"comment": "At $t=t_Q([7,6,2])$, $C_t$ has CM order discriminant $\\Delta=-20$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 58."
},
"cm-m84-15-6-2": {
"param-latex": "$t_Q([15,6,2])$",
"comment": "At $t=t_Q([15,6,2])$, $C_t$ has CM order discriminant $\\Delta=-84$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 60."
},
"cm-m132-21-6-2": {
"param-latex": "$t_Q([21,6,2])$",
"comment": "At $t=t_Q([21,6,2])$, $C_t$ has CM order discriminant $\\Delta=-132$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 61."
},
"cm-m195-21-15-5": {
"param-latex": "$t_Q([21,15,5])$",
"comment": "At $t=t_Q([21,15,5])$, $C_t$ has CM order discriminant $\\Delta=-195$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 63."
},
"cm-m195-51-3-1": {
"param-latex": "$t_Q([51,3,1])$",
"comment": "At $t=t_Q([51,3,1])$, $C_t$ has CM order discriminant $\\Delta=-195$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 63."
},
"cm-m228-33-6-2": {
"param-latex": "$t_Q([33,6,2])$",
"comment": "At $t=t_Q([33,6,2])$, $C_t$ has CM order discriminant $\\Delta=-228$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 66."
},
"cm-m315-27-21-7": {
"param-latex": "$t_Q([27,21,7])$",
"comment": "At $t=t_Q([27,21,7])$, $C_t$ has CM order discriminant $\\Delta=-315$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 67."
},
"cm-m315-27-15-5": {
"param-latex": "$t_Q([27,15,5])$",
"comment": "At $t=t_Q([27,15,5])$, $C_t$ has CM order discriminant $\\Delta=-315$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 67."
},
"cm-m315-81-3-1": {
"param-latex": "$t_Q([81,3,1])$",
"comment": "At $t=t_Q([81,3,1])$, $C_t$ has CM order discriminant $\\Delta=-315$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 67."
},
"cm-m35-11-3-1": {
"param-latex": "$t_Q([11,3,1])$",
"comment": "At $t=t_Q([11,3,1])$, $C_t$ has CM order discriminant $\\Delta=-35$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 68."
},
"cm-m435-33-15-5": {
"param-latex": "$t_Q([33,15,5])$",
"comment": "At $t=t_Q([33,15,5])$, $C_t$ has CM order discriminant $\\Delta=-435$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 69."
},
"cm-m435-111-3-1": {
"param-latex": "$t_Q([111,3,1])$",
"comment": "At $t=t_Q([111,3,1])$, $C_t$ has CM order discriminant $\\Delta=-435$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 69."
},
"cm-m372-51-6-2": {
"param-latex": "$t_Q([51,6,2])$",
"comment": "At $t=t_Q([51,6,2])$, $C_t$ has CM order discriminant $\\Delta=-372$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 70."
},
"cm-m555-39-15-5": {
"param-latex": "$t_Q([39,15,5])$",
"comment": "At $t=t_Q([39,15,5])$, $C_t$ has CM order discriminant $\\Delta=-555$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 71."
},
"cm-m555-141-3-1": {
"param-latex": "$t_Q([141,3,1])$",
"comment": "At $t=t_Q([141,3,1])$, $C_t$ has CM order discriminant $\\Delta=-555$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 71."
},
"cm-m795-51-15-5": {
"param-latex": "$t_Q([51,15,5])$",
"comment": "At $t=t_Q([51,15,5])$, $C_t$ has CM order discriminant $\\Delta=-795$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 72."
},
"cm-m795-201-3-1": {
"param-latex": "$t_Q([201,3,1])$",
"comment": "At $t=t_Q([201,3,1])$, $C_t$ has CM order discriminant $\\Delta=-795$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 72."
},
"cm-m708-93-6-2": {
"param-latex": "$t_Q([93,6,2])$",
"comment": "At $t=t_Q([93,6,2])$, $C_t$ has CM order discriminant $\\Delta=-708$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 73."
},
"cm-m483-33-21-7": {
"param-latex": "$t_Q([33,21,7])$",
"comment": "At $t=t_Q([33,21,7])$, $C_t$ has CM order discriminant $\\Delta=-483$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 74."
},
"cm-m483-123-3-1": {
"param-latex": "$t_Q([123,3,1])$",
"comment": "At $t=t_Q([123,3,1])$, $C_t$ has CM order discriminant $\\Delta=-483$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 74."
},
"cm-m627-39-33-11": {
"param-latex": "$t_Q([39,33,11])$",
"comment": "At $t=t_Q([39,33,11])$, $C_t$ has CM order discriminant $\\Delta=-627$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 75."
},
"cm-m627-159-3-1": {
"param-latex": "$t_Q([159,3,1])$",
"comment": "At $t=t_Q([159,3,1])$, $C_t$ has CM order discriminant $\\Delta=-627$. Its parameter field $\\mathbb Q(t)$ is totally real and biquadratic. CITE{ST}, Table 2, row 75."
},
"cm-m7-4-3-1": {
"param-latex": "$t_Q([4,3,1])$",
"comment": "At $t=t_Q([4,3,1])$, $C_t$ has CM order discriminant $\\Delta=-7$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-1-2."
},
"cm-m19-7-3-1": {
"param-latex": "$t_Q([7,3,1])$",
"comment": "At $t=t_Q([7,3,1])$, $C_t$ has CM order discriminant $\\Delta=-19$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-2-2."
},
"cm-m43-13-3-1": {
"param-latex": "$t_Q([13,3,1])$",
"comment": "At $t=t_Q([13,3,1])$, $C_t$ has CM order discriminant $\\Delta=-43$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-4-2."
},
"cm-m63-18-3-1": {
"param-latex": "$t_Q([18,3,1])$",
"comment": "At $t=t_Q([18,3,1])$, $C_t$ has CM order discriminant $\\Delta=-63$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-5-2."
},
"cm-m67-19-3-1": {
"param-latex": "$t_Q([19,3,1])$",
"comment": "At $t=t_Q([19,3,1])$, $C_t$ has CM order discriminant $\\Delta=-67$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-6-2."
},
"cm-m163-43-3-1": {
"param-latex": "$t_Q([43,3,1])$",
"comment": "At $t=t_Q([43,3,1])$, $C_t$ has CM order discriminant $\\Delta=-163$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-7-2."
},
"cm-m171-45-3-1": {
"param-latex": "$t_Q([45,3,1])$",
"comment": "At $t=t_Q([45,3,1])$, $C_t$ has CM order discriminant $\\Delta=-171$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-8-2."
},
"cm-m387-99-3-1": {
"param-latex": "$t_Q([99,3,1])$",
"comment": "At $t=t_Q([99,3,1])$, $C_t$ has CM order discriminant $\\Delta=-387$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-9-2."
},
"cm-m603-153-3-1": {
"param-latex": "$t_Q([153,3,1])$",
"comment": "At $t=t_Q([153,3,1])$, $C_t$ has CM order discriminant $\\Delta=-603$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-13-2."
},
"cm-m1467-369-3-1": {
"param-latex": "$t_Q([369,3,1])$",
"comment": "At $t=t_Q([369,3,1])$, $C_t$ has CM order discriminant $\\Delta=-1467$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-14-2."
},
"cm-m39-12-3-1": {
"param-latex": "$t_Q([12,3,1])$",
"comment": "At $t=t_Q([12,3,1])$, $C_t$ has CM order discriminant $\\Delta=-39$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-18-2."
},
"cm-m219-57-3-1": {
"param-latex": "$t_Q([57,3,1])$",
"comment": "At $t=t_Q([57,3,1])$, $C_t$ has CM order discriminant $\\Delta=-219$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-19-2."
},
"cm-m291-75-3-1": {
"param-latex": "$t_Q([75,3,1])$",
"comment": "At $t=t_Q([75,3,1])$, $C_t$ has CM order discriminant $\\Delta=-291$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-20-2."
},
"cm-m363-93-3-1": {
"param-latex": "$t_Q([93,3,1])$",
"comment": "At $t=t_Q([93,3,1])$, $C_t$ has CM order discriminant $\\Delta=-363$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-21-2."
},
"cm-m507-129-3-1": {
"param-latex": "$t_Q([129,3,1])$",
"comment": "At $t=t_Q([129,3,1])$, $C_t$ has CM order discriminant $\\Delta=-507$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-22-2."
},
"cm-m723-183-3-1": {
"param-latex": "$t_Q([183,3,1])$",
"comment": "At $t=t_Q([183,3,1])$, $C_t$ has CM order discriminant $\\Delta=-723$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-23-2."
},
"cm-m1227-309-3-1": {
"param-latex": "$t_Q([309,3,1])$",
"comment": "At $t=t_Q([309,3,1])$, $C_t$ has CM order discriminant $\\Delta=-1227$. Its quartic parameter field $\\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. CITE{Supplement}, entry ν-N4-24-2."
}
}