Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-219185292x-1268107303664\)
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(homogenize, simplify) |
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\(y^2z=x^3-219185292xz^2-1268107303664z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-219185292x-1268107303664\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(18050, 810144\right) \) | $6.5636621499649224557073697434$ | $\infty$ |
| \( \left(157890, 62451616\right) \) | $9.7697847542828064571509263651$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([18050:810144:1]\) | $6.5636621499649224557073697434$ | $\infty$ |
| \([157890:62451616:1]\) | $9.7697847542828064571509263651$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(18050, 810144\right) \) | $6.5636621499649224557073697434$ | $\infty$ |
| \( \left(157890, 62451616\right) \) | $9.7697847542828064571509263651$ | $\infty$ |
Integral points
\((18050,\pm 810144)\), \((157890,\pm 62451616)\)
\([18050:\pm 810144:1]\), \([157890:\pm 62451616:1]\)
\((18050,\pm 810144)\), \((157890,\pm 62451616)\)
Invariants
| Conductor: | $N$ | = | \( 480960 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 167$ |
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| Minimal Discriminant: | $\Delta$ | = | $-20768446065882571737661440$ | = | $-1 \cdot 2^{19} \cdot 3^{7} \cdot 5 \cdot 167^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{6093832136609347161121}{108676727597808690} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 5^{-1} \cdot 19^{3} \cdot 167^{-7} \cdot 961339^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.6547199337037616836834120962$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.0656930185297888738599412956$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9971440792413382$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.29369283662851$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $57.926504191339591881290151549$ |
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| Real period: | $\Omega$ | ≈ | $0.019597792547161316646198286998$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $18.163705793986301201588924137 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 18.163705794 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.019598 \cdot 57.926504 \cdot 16}{1^2} \\ & \approx 18.163705794\end{aligned}$$
Modular invariants
Modular form 480960.2.a.eq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 101154816 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{9}^{*}$ | additive | -1 | 6 | 19 | 1 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $167$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
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 |
|---|---|---|---|
| $7$ | 7B.6.3 | 7.24.0.2 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 140280 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 167 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 140272 & 140273 \\ 70147 & 6 \end{array}\right),\left(\begin{array}{rr} 117608 & 7 \\ 107513 & 140274 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 140267 & 14 \\ 140266 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 140272 & 140273 \\ 105217 & 6 \end{array}\right),\left(\begin{array}{rr} 35073 & 100208 \\ 70126 & 115193 \end{array}\right),\left(\begin{array}{rr} 84176 & 7 \\ 84161 & 140274 \end{array}\right),\left(\begin{array}{rr} 93512 & 140273 \\ 93527 & 6 \end{array}\right)$.
The torsion field $K:=\Q(E[140280])$ is a degree-$574559373376880640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/140280\Z)$.
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 | $4$ | \( 7515 = 3^{2} \cdot 5 \cdot 167 \) |
| $3$ | additive | $8$ | \( 53440 = 2^{6} \cdot 5 \cdot 167 \) |
| $5$ | split multiplicative | $6$ | \( 96192 = 2^{6} \cdot 3^{2} \cdot 167 \) |
| $7$ | good | $2$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
| $167$ | nonsplit multiplicative | $168$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 480960eq
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 5010h2, 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.