Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-707763x+222280562\)
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(homogenize, simplify) |
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\(y^2z=x^3-707763xz^2+222280562z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-707763x+222280562\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(7, 14742)$ | $1.2636865896410385852268724621$ | $\infty$ |
| $(574, 2268)$ | $2.7425352546315841130468812559$ | $\infty$ |
| $(553, 0)$ | $0$ | $2$ |
Integral points
\((-343,\pm 20608)\), \((-97,\pm 17030)\), \((7,\pm 14742)\), \((169,\pm 10368)\), \((319,\pm 5382)\), \( \left(553, 0\right) \), \((562,\pm 1422)\), \((574,\pm 2268)\), \((10537,\pm 1078272)\), \((23849,\pm 3680768)\)
Invariants
| Conductor: | $N$ | = | \( 458640 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $1345936444489728000$ | = | $2^{20} \cdot 3^{11} \cdot 5^{3} \cdot 7^{3} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{38282975119927}{1314144000} \) | = | $2^{-8} \cdot 3^{-5} \cdot 5^{-3} \cdot 13^{-2} \cdot 33703^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2504144268236310479089067203$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.52148356466580256651771379452$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9485665982496478$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9907267159040702$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.3077154174660258846382648047$ |
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| Real period: | $\Omega$ | ≈ | $0.26917373696263757277057140444$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $14.245601915652175701217267373 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.245601916 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.269174 \cdot 3.307715 \cdot 64}{2^2} \\ & \approx 14.245601916\end{aligned}$$
Modular invariants
Modular form 458640.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5898240 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{12}^{*}$ | additive | -1 | 4 | 20 | 8 |
| $3$ | $4$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $7$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $13$ | $2$ | $I_{2}$ | nonsplit 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5460 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1369 & 4096 \\ 1364 & 4095 \end{array}\right),\left(\begin{array}{rr} 1094 & 1 \\ 2183 & 0 \end{array}\right),\left(\begin{array}{rr} 1822 & 1 \\ 1819 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3904 & 1 \\ 3119 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 5457 & 4 \\ 5456 & 5 \end{array}\right),\left(\begin{array}{rr} 4201 & 4 \\ 2942 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[5460])$ is a degree-$9738607656960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5460\Z)$.
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 | $2$ | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| $3$ | additive | $8$ | \( 10192 = 2^{4} \cdot 7^{2} \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 91728 = 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $20$ | \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 35280 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 458640s
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 19110dd1, its twist by $12$.
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.