Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-4051308x-3077600848\)
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(homogenize, simplify) |
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\(y^2z=x^3-4051308xz^2-3077600848z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4051308x-3077600848\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(10933, 1122255\right) \) | $3.1080214584199915573975108418$ | $\infty$ |
| \( \left(-1292, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([10933:1122255:1]\) | $3.1080214584199915573975108418$ | $\infty$ |
| \([-1292:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(10933, 1122255\right) \) | $3.1080214584199915573975108418$ | $\infty$ |
| \( \left(-1292, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-1292, 0\right) \), \((10933,\pm 1122255)\)
\([-1292:0:1]\), \([10933:\pm 1122255:1]\)
\( \left(-1292, 0\right) \), \((10933,\pm 1122255)\)
Invariants
| Conductor: | $N$ | = | \( 469440 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 163$ |
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| Minimal Discriminant: | $\Delta$ | = | $163905733559408394240$ | = | $2^{18} \cdot 3^{11} \cdot 5 \cdot 163^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{38480618749557529}{857682789615} \) | = | $3^{-5} \cdot 5^{-1} \cdot 163^{-4} \cdot 337609^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6662728754609689201940485348$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0772459602869961103705777342$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9476298241848735$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.384405949110108$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.1080214584199915573975108418$ |
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| Real period: | $\Omega$ | ≈ | $0.10655894283656456615838922049$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.2989996947614706396687265240 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.298999695 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.106559 \cdot 3.108021 \cdot 64}{2^2} \\ & \approx 5.298999695\end{aligned}$$
Modular invariants
Modular form 469440.2.a.bj
For more coefficients, see the Downloads section to the right.
| Modular degree: | 15728640 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{8}^{*}$ | additive | 1 | 6 | 18 | 0 |
| $3$ | $4$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $163$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 | 8.12.0.7 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 19560 = 2^{3} \cdot 3 \cdot 5 \cdot 163 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 19553 & 8 \\ 19552 & 9 \end{array}\right),\left(\begin{array}{rr} 2448 & 12233 \\ 2473 & 2520 \end{array}\right),\left(\begin{array}{rr} 13036 & 19559 \\ 6497 & 19554 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 19554 & 19555 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 8641 & 8 \\ 15004 & 33 \end{array}\right),\left(\begin{array}{rr} 12221 & 12222 \\ 17102 & 7325 \end{array}\right),\left(\begin{array}{rr} 15656 & 3 \\ 5 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[19560])$ is a degree-$517242181386240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/19560\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 45 = 3^{2} \cdot 5 \) |
| $3$ | additive | $8$ | \( 52160 = 2^{6} \cdot 5 \cdot 163 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 93888 = 2^{6} \cdot 3^{2} \cdot 163 \) |
| $163$ | split multiplicative | $164$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 469440.bj
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2445.c1, its twist by $-24$.
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.