Properties

Label 2.2.40.1-90.1-f11
Base field \(\Q(\sqrt{10}) \)
Conductor norm \( 90 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{10}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 10 \); class number \(2\).

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-10, 0, 1]))
 
Copy content gp:K = nfinit(Polrev([-10, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10, 0, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}-5334{x}-150368\)
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-5334,0]),K([-150368,0])])
 
Copy content gp:E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,0]),Polrev([-5334,0]),Polrev([-150368,0])], K);
 
Copy content magma:E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-5334,0],K![-150368,0]]);
 

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

\(\Z/{2}\Z \oplus \Z/{2}\Z\)

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-\frac{169}{4} : \frac{165}{8} : 1\right)$$0$$2$
$\left(-20 a + 21 : 10 a - 11 : 1\right)$$0$$2$

Invariants

Conductor: $\frak{N}$ = \((-3a)\) = \((2,a)\cdot(3,a+1)\cdot(3,a+2)\cdot(5,a)\)
Copy content comment:Compute the conductor
 
Copy content sage:E.conductor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 90 \) = \(2\cdot3\cdot3\cdot5\)
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: $\Delta$ = $81000$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((81000)\) = \((2,a)^{6}\cdot(3,a+1)^{4}\cdot(3,a+2)^{4}\cdot(5,a)^{6}\)
Copy content comment:Compute the discriminant
 
Copy content sage:E.discriminant()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 6561000000 \) = \(2^{6}\cdot3^{4}\cdot3^{4}\cdot5^{6}\)
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: $j$ = \( \frac{16778985534208729}{81000} \)
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: $\mathrm{End}(E)$ = \(\Z\)   
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$ = \(\Z\)    (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: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$= \( 0 \)
Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content magma:Rank(E);
 
Mordell-Weil rank: $r$ = \(0\)
Regulator: $\mathrm{Reg}(E/K)$ = \( 1 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ = \( 1 \)
Global period: $\Omega(E/K)$ \( 0.31209880923978050798954611509762644805 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 64 \)  =  \(2\cdot2^{2}\cdot2^{2}\cdot2\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(4\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 1.7764975660033988509252275712977984037 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 9 \) (rounded)

BSD formula

$$\begin{aligned}1.776497566 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 9 \cdot 0.312099 \cdot 1 \cdot 64 } { {4^2 \cdot 6.324555} } \\ & \approx 1.776497566 \end{aligned}$$

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 semistable. There are 4 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((2,a)\) \(2\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((3,a+1)\) \(3\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((3,a+2)\) \(3\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((5,a)\) \(5\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 90.1-f consists of curves linked by isogenies of degrees dividing 24.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 30.a1
\(\Q\) 4800.cq1