Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-7131x-271757\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-7131xz^2-271757z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-9241155x-12651359490\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(131, 957)$ | $3.0608901961359613132948510806$ | $\infty$ |
Integral points
\( \left(131, 957\right) \), \( \left(131, -1089\right) \)
Invariants
| Conductor: | $N$ | = | \( 444675 \) | = | $3 \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $-8561603028825$ | = | $-1 \cdot 3^{7} \cdot 5^{2} \cdot 7^{6} \cdot 11^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{10241915}{2187} \) | = | $-1 \cdot 3^{-7} \cdot 5 \cdot 127^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2028141868502848217034016776$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.63785435794931452929822047748$ |
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| $abc$ quality: | $Q$ | ≈ | $0.931175213655602$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.963767754894653$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.0608901961359613132948510806$ |
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| Real period: | $\Omega$ | ≈ | $0.25683459665701935895402897654$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 14 $ = $ 7\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.005994985104062992518873621 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.005994985 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.256835 \cdot 3.060890 \cdot 14}{1^2} \\ & \approx 11.005994985\end{aligned}$$
Modular invariants
Modular form 444675.2.a.gj
For more coefficients, see the Downloads section to the right.
| Modular degree: | 762048 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $5$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
| $7$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 132 = 2^{2} \cdot 3 \cdot 11 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 89 & 2 \\ 89 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 67 & 2 \\ 67 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 131 & 2 \\ 130 & 3 \end{array}\right),\left(\begin{array}{rr} 13 & 2 \\ 13 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 131 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[132])$ is a degree-$30412800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/132\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | good | $2$ | \( 40425 = 3 \cdot 5^{2} \cdot 7^{2} \cdot 11 \) |
| $3$ | split multiplicative | $4$ | \( 148225 = 5^{2} \cdot 7^{2} \cdot 11^{2} \) |
| $5$ | additive | $10$ | \( 17787 = 3 \cdot 7^{2} \cdot 11^{2} \) |
| $7$ | additive | $26$ | \( 3025 = 5^{2} \cdot 11^{2} \) |
| $11$ | additive | $42$ | \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 444675.gj consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 9075.m1, its twist by $-7$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.