Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-23452x-1111860\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-23452xz^2-1111860z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1899639x-804847050\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-117, 210)$ | $3.9512580856310126907654808738$ | $\infty$ |
| $(-54, 0)$ | $0$ | $2$ |
Integral points
\((-117,\pm 210)\), \( \left(-54, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 414736 \) | = | $2^{4} \cdot 7^{2} \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $298970912850176$ | = | $2^{8} \cdot 7^{3} \cdot 23^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{109744}{23} \) | = | $2^{4} \cdot 19^{3} \cdot 23^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4924450395448798423139165684$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0238777260568202023106194477$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7730187674526566$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.231580552619347$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.9512580856310126907654808738$ |
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| Real period: | $\Omega$ | ≈ | $0.39149810938783239011478281106$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.0938201404558548150006486897 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.093820140 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.391498 \cdot 3.951258 \cdot 8}{2^2} \\ & \approx 3.093820140\end{aligned}$$
Modular invariants
Modular form 414736.2.a.j
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1622016 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | 1 | 4 | 8 | 0 |
| $7$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $23$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
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 \( 644 = 2^{2} \cdot 7 \cdot 23 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 641 & 4 \\ 640 & 5 \end{array}\right),\left(\begin{array}{rr} 562 & 1 \\ 363 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 96 & 1 \\ 459 & 0 \end{array}\right),\left(\begin{array}{rr} 485 & 162 \\ 160 & 483 \end{array}\right)$.
The torsion field $K:=\Q(E[644])$ is a degree-$4308885504$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/644\Z)$.
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 | $2$ | \( 3703 = 7 \cdot 23^{2} \) |
| $7$ | additive | $20$ | \( 8464 = 2^{4} \cdot 23^{2} \) |
| $23$ | additive | $288$ | \( 784 = 2^{4} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 414736j
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 9016c1, its twist by $92$.
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.