Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+16317x+231182\)
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(homogenize, simplify) |
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\(y^2z=x^3+16317xz^2+231182z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+16317x+231182\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(7, 588\right) \) | $0.90903141062359909995613395757$ | $\infty$ |
| \( \left(-14, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([7:588:1]\) | $0.90903141062359909995613395757$ | $\infty$ |
| \([-14:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(7, 588\right) \) | $0.90903141062359909995613395757$ | $\infty$ |
| \( \left(-14, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-14, 0\right) \), \((7,\pm 588)\), \((182,\pm 3038)\), \((791,\pm 22540)\)
\([-14:0:1]\), \([7:\pm 588:1]\), \([182:\pm 3038:1]\), \([791:\pm 22540:1]\)
\( \left(-14, 0\right) \), \((7,\pm 588)\), \((182,\pm 3038)\), \((791,\pm 22540)\)
Invariants
| Conductor: | $N$ | = | \( 405720 \) | = | $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 23$ |
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| Minimal Discriminant: | $\Delta$ | = | $-301124215526400$ | = | $-1 \cdot 2^{10} \cdot 3^{3} \cdot 5^{2} \cdot 7^{7} \cdot 23^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{147704148}{92575} \) | = | $2^{2} \cdot 3^{6} \cdot 5^{-2} \cdot 7^{-1} \cdot 23^{-2} \cdot 37^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4672717081587834835358262651$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.35795908900252168304668818373$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7931605637068334$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.1528050650241064$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.90903141062359909995613395757$ |
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| Real period: | $\Omega$ | ≈ | $0.33839711309295180622668279084$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.9218176810534332260609129211 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.921817681 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.338397 \cdot 0.909031 \cdot 64}{2^2} \\ & \approx 4.921817681\end{aligned}$$
Modular invariants
Modular form 405720.2.a.k
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1474560 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | 1 | 3 | 10 | 0 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $23$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 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 | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9660 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6721 & 4 \\ 3782 & 9 \end{array}\right),\left(\begin{array}{rr} 3224 & 1 \\ 6439 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 5797 & 4 \\ 1934 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 9657 & 4 \\ 9656 & 5 \end{array}\right),\left(\begin{array}{rr} 5522 & 1 \\ 6899 & 0 \end{array}\right),\left(\begin{array}{rr} 2416 & 7249 \\ 7245 & 2416 \end{array}\right)$.
The torsion field $K:=\Q(E[9660])$ is a degree-$99276722012160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9660\Z)$.
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 | $2$ | \( 147 = 3 \cdot 7^{2} \) |
| $3$ | additive | $6$ | \( 45080 = 2^{3} \cdot 5 \cdot 7^{2} \cdot 23 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 81144 = 2^{3} \cdot 3^{2} \cdot 7^{2} \cdot 23 \) |
| $7$ | additive | $32$ | \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 17640 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 405720k
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 57960bg2, its twist by $21$.
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.