Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-73413x-7633683\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-73413xz^2-7633683z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-95143275x-355871684250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(342, 2529)$ | $3.1567544841843082400189497553$ | $\infty$ |
| $(-657/4, 657/8)$ | $0$ | $2$ |
Integral points
\( \left(342, 2529\right) \), \( \left(342, -2871\right) \)
Invariants
| Conductor: | $N$ | = | \( 458850 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 19 \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $188420443312500$ | = | $2^{2} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 19 \cdot 23^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{2800418713303177}{12058908372} \) | = | $2^{-2} \cdot 3^{-4} \cdot 7^{-1} \cdot 19^{-1} \cdot 23^{-4} \cdot 47^{3} \cdot 2999^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5926471409497462797958853620$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.78792818473269609249550569539$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9015119953724396$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.469124912558644$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.1567544841843082400189497553$ |
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| Real period: | $\Omega$ | ≈ | $0.29011407227482431019693773597$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2^{2}\cdot2\cdot1\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $14.653102377256354169473494027 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.653102377 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.290114 \cdot 3.156754 \cdot 64}{2^2} \\ & \approx 14.653102377\end{aligned}$$
Modular invariants
Modular form 458850.2.a.eu
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2359296 |
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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 6 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $23$ | $4$ | $I_{4}$ | 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 367080 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 284491 & 284490 \\ 192730 & 302851 \end{array}\right),\left(\begin{array}{rr} 335161 & 146840 \\ 312820 & 220281 \end{array}\right),\left(\begin{array}{rr} 174371 & 247780 \\ 174410 & 137661 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 46376 & 146835 \\ 212525 & 293666 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 10496 & 146835 \\ 52445 & 293666 \end{array}\right),\left(\begin{array}{rr} 244721 & 146840 \\ 318140 & 220281 \end{array}\right),\left(\begin{array}{rr} 367073 & 8 \\ 367072 & 9 \end{array}\right),\left(\begin{array}{rr} 293663 & 0 \\ 0 & 367079 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 367074 & 367075 \end{array}\right)$.
The torsion field $K:=\Q(E[367080])$ is a degree-$48891800056548556800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/367080\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 3325 = 5^{2} \cdot 7 \cdot 19 \) |
| $3$ | split multiplicative | $4$ | \( 152950 = 2 \cdot 5^{2} \cdot 7 \cdot 19 \cdot 23 \) |
| $5$ | additive | $14$ | \( 18354 = 2 \cdot 3 \cdot 7 \cdot 19 \cdot 23 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 65550 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \cdot 23 \) |
| $19$ | split multiplicative | $20$ | \( 24150 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 19950 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 458850.eu
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 18354.c1, its twist by $5$.
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.