Properties

Label 2.0.23.1-108.6-b6
Base field Q(23)\Q(\sqrt{-23})
Conductor norm 108 108
CM no
Base change no
Q-curve yes
Torsion order 4 4
Rank 1 1

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Base field Q(23)\Q(\sqrt{-23})

Generator aa, with minimal polynomial x2x+6 x^{2} - x + 6 ; class number 33.

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([6, -1, 1]))
 
Copy content gp:K = nfinit(Polrev([6, -1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6, -1, 1]);
 

Weierstrass equation

y2+xy+y=x3+(a1)x2+(3a+10)x+a+7{y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+\left(-3a+10\right){x}+a+7
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([1,0]),K([-1,-1]),K([1,0]),K([10,-3]),K([7,1])])
 
Copy content gp:E = ellinit([Polrev([1,0]),Polrev([-1,-1]),Polrev([1,0]),Polrev([10,-3]),Polrev([7,1])], K);
 
Copy content magma:E := EllipticCurve([K![1,0],K![-1,-1],K![1,0],K![10,-3],K![7,1]]);
 

This is a global minimal model.

Copy content comment:Test whether it is a global minimal model
 
Copy content sage:E.is_global_minimal_model()
 

Mordell-Weil group structure

ZZ/2ZZ/2Z\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(1:a+1:1)\left(-1 : a + 1 : 1\right)0.273865281215312181433343923119408506380.27386528121531218143334392311940850638\infty
(34a+14:38a58:1)\left(-\frac{3}{4} a + \frac{1}{4} : \frac{3}{8} a - \frac{5}{8} : 1\right)0022
(14a12:18a14:1)\left(-\frac{1}{4} a - \frac{1}{2} : \frac{1}{8} a - \frac{1}{4} : 1\right)0022

Invariants

Conductor: N\frak{N} = (18,6a)(18,6a) = (2,a)(2,a+1)(3,a)2(3,a+2)(2,a)\cdot(2,a+1)\cdot(3,a)^{2}\cdot(3,a+2)
Copy content comment:Compute the conductor
 
Copy content sage:E.conductor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Conductor norm: N(N)N(\frak{N}) = 108 108 = 223232\cdot2\cdot3^{2}\cdot3
Copy content comment:Compute the norm of the conductor
 
Copy content sage:E.conductor().norm()
 
Copy content gp:idealnorm(ellglobalred(E)[1])
 
Copy content magma:Norm(Conductor(E));
 
Discriminant: Δ\Delta = 13104a4104013104a-41040
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (13104a41040)(13104a-41040) = (2,a)4(2,a+1)8(3,a)10(3,a+2)2(2,a)^{4}\cdot(2,a+1)^{8}\cdot(3,a)^{10}\cdot(3,a+2)^{2}
Copy content comment:Compute the discriminant
 
Copy content sage:E.discriminant()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Discriminant norm: N(Dmin)=N(Δ)N(\frak{D}_{\mathrm{min}}) = N(\Delta) = 2176782336 2176782336 = 2428310322^{4}\cdot2^{8}\cdot3^{10}\cdot3^{2}
Copy content comment:Compute the norm of the discriminant
 
Copy content sage:E.discriminant().norm()
 
Copy content gp:norm(E.disc)
 
Copy content magma:Norm(Discriminant(E));
 
j-invariant: jj = 5425626720736a+2257607910368 -\frac{54256267}{20736} a + \frac{22576079}{10368}
Copy content comment:Compute the j-invariant
 
Copy content sage:E.j_invariant()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z\Z   
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}}) = Z\Z    (no potential complex multiplication)
Copy content comment:Test for Complex Multiplication
 
Copy content sage:E.has_cm(), E.cm_discriminant()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)

BSD invariants

Analytic rank: ranr_{\mathrm{an}}= 1 1
Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content magma:Rank(E);
 
Mordell-Weil rank: rr = 11
Regulator: Reg(E/K)\mathrm{Reg}(E/K) 0.27386528121531218143334392311940850638 0.27386528121531218143334392311940850638
Néron-Tate Regulator: RegNT(E/K)\mathrm{Reg}_{\mathrm{NT}}(E/K) 0.547730562430624362866687846238817012760 0.547730562430624362866687846238817012760
Global period: Ω(E/K)\Omega(E/K) 4.2631400829787424910520982906442720882 4.2631400829787424910520982906442720882
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 64 64  =  2222222^{2}\cdot2\cdot2^{2}\cdot2
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 44
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! 1.9475680945168386727622166971056384619 1.9475680945168386727622166971056384619
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

1.947568095L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/214.2631400.54773164424.7958321.947568095\begin{aligned}1.947568095 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 4.263140 \cdot 0.547731 \cdot 64 } { {4^2 \cdot 4.795832} } \\ & \approx 1.947568095 \end{aligned}

Local data at primes of bad reduction

Copy content comment:Compute the local reduction data at primes of bad reduction
 
Copy content sage:E.local_data()
 
Copy content magma:LocalInformation(E);
 

This elliptic curve is not semistable. There are 4 primes p\frak{p} of bad reduction.

p\mathfrak{p} N(p)N(\mathfrak{p}) Tamagawa number Kodaira symbol Reduction type Root number ordp(N\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}) ordp(Dmin\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}) ordp(den(j))\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))
(2,a)(2,a) 22 44 I4I_{4} Split multiplicative 1-1 11 44 44
(2,a+1)(2,a+1) 22 22 I8I_{8} Non-split multiplicative 11 11 88 88
(3,a)(3,a) 33 44 I4I_{4}^{*} Additive 1-1 22 1010 44
(3,a+2)(3,a+2) 33 22 I2I_{2} Non-split multiplicative 11 11 22 22

Galois Representations

The mod p p Galois Representation has maximal image for all primes p<1000 p < 1000 except those listed.

prime Image of Galois Representation
22 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2 and 4.
Its isogeny class 108.6-b consists of curves linked by isogenies of degrees dividing 8.

Base change

This elliptic curve is a Q\Q-curve.

It is not the base change of an elliptic curve defined over any subfield.