Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-18461096586x+965455066178280\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-18461096586xz^2+965455066178280z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-295377545379x+61788828857864542\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{314595}{4}, -\frac{314595}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([629190:-314595:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(314594, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 466578 \) | = | $2 \cdot 3^{2} \cdot 7^{2} \cdot 23^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $8095367397239575658444728188$ | = | $2^{2} \cdot 3^{22} \cdot 7^{7} \cdot 23^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{54804145548726848737}{637608031452} \) | = | $2^{-2} \cdot 3^{-16} \cdot 7^{-1} \cdot 23^{-2} \cdot 193^{3} \cdot 19681^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.5059828798661240550786669881$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4159745530398377114249915820$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0156853826442096$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.322627463270344$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.037659756406173925515194717239$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.7115024612445226370940196412 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.711502461 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.037660 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 2.711502461\end{aligned}$$
Modular invariants
Modular form 466578.2.a.ce
For more coefficients, see the Downloads section to the right.
| Modular degree: | 830472192 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 4 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $4$ | $I_{16}^{*}$ | additive | -1 | 2 | 22 | 16 |
| $7$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $23$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 16.24.0.13 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 7724 & 7725 \end{array}\right),\left(\begin{array}{rr} 7713 & 16 \\ 7712 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 2592 \\ 2964 & 2905 \end{array}\right),\left(\begin{array}{rr} 5363 & 5136 \\ 7536 & 2891 \end{array}\right),\left(\begin{array}{rr} 2192 & 2571 \\ 4461 & 14 \end{array}\right),\left(\begin{array}{rr} 2575 & 0 \\ 0 & 7727 \end{array}\right),\left(\begin{array}{rr} 7414 & 3225 \\ 2901 & 6442 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 7630 & 7715 \end{array}\right)$.
The torsion field $K:=\Q(E[7728])$ is a degree-$3309224067072$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7728\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 233289 = 3^{2} \cdot 7^{2} \cdot 23^{2} \) |
| $3$ | additive | $8$ | \( 51842 = 2 \cdot 7^{2} \cdot 23^{2} \) |
| $7$ | additive | $32$ | \( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \) |
| $23$ | additive | $288$ | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 466578ce
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 966g5, its twist by $-483$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.