Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-174367x-28082890\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-174367xz^2-28082890z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-14123754x-20430055575\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-237936463/984064, 11748791145/976191488)$ | $16.434188486213623750197027829$ | $\infty$ |
| $(-242, 0)$ | $0$ | $2$ |
Integral points
\( \left(-242, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 464232 \) | = | $2^{3} \cdot 3 \cdot 23 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $5910164517456$ | = | $2^{4} \cdot 3^{3} \cdot 23 \cdot 29^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{61604313088}{621} \) | = | $2^{11} \cdot 3^{-3} \cdot 23^{-1} \cdot 311^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6095717682821496585910406099$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.30512520689773579147300611343$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0662343776302834$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.664917950100567$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $16.434188486213623750197027829$ |
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| Real period: | $\Omega$ | ≈ | $0.23363265047244178996439868016$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot3\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.518689043193324423701636681 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.518689043 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.233633 \cdot 16.434188 \cdot 12}{2^2} \\ & \approx 11.518689043\end{aligned}$$
Modular invariants
Modular form 464232.2.a.r
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2408448 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | 1 | 3 | 4 | 0 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $23$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $29$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 \( 16008 = 2^{3} \cdot 3 \cdot 23 \cdot 29 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 15455 & 0 \\ 0 & 16007 \end{array}\right),\left(\begin{array}{rr} 1480 & 14355 \\ 2581 & 15458 \end{array}\right),\left(\begin{array}{rr} 10876 & 15457 \\ 3335 & 13254 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 16002 & 16003 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 8353 & 8352 \\ 10846 & 5047 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 16001 & 8 \\ 16000 & 9 \end{array}\right),\left(\begin{array}{rr} 12355 & 12354 \\ 2842 & 13051 \end{array}\right)$.
The torsion field $K:=\Q(E[16008])$ is a degree-$279905202339840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/16008\Z)$.
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 | $2$ | \( 58029 = 3 \cdot 23 \cdot 29^{2} \) |
| $3$ | split multiplicative | $4$ | \( 154744 = 2^{3} \cdot 23 \cdot 29^{2} \) |
| $23$ | split multiplicative | $24$ | \( 20184 = 2^{3} \cdot 3 \cdot 29^{2} \) |
| $29$ | additive | $422$ | \( 552 = 2^{3} \cdot 3 \cdot 23 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 464232.r
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 552.c3, its twist by $29$.
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.