Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-146935x-17919514\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-146935xz^2-17919514z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-190427139x-835481552130\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-191, 1880)$ | $3.1426885811840015800290559147$ | $\infty$ |
| $(-293, 146)$ | $0$ | $2$ |
| $(-141, 70)$ | $0$ | $2$ |
Integral points
\( \left(-293, 146\right) \), \( \left(-191, 1880\right) \), \( \left(-191, -1690\right) \), \( \left(-141, 70\right) \), \( \left(790, 18557\right) \), \( \left(790, -19348\right) \)
Invariants
| Conductor: | $N$ | = | \( 405042 \) | = | $2 \cdot 3 \cdot 11 \cdot 17 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $64496359167059556$ | = | $2^{2} \cdot 3^{4} \cdot 11^{4} \cdot 17^{2} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{7457162887153}{1370924676} \) | = | $2^{-2} \cdot 3^{-4} \cdot 7^{3} \cdot 11^{-4} \cdot 17^{-2} \cdot 2791^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9439052558357914403488378143$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.47168576625257121034432409836$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9400064681728089$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.663860012429093$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.1426885811840015800290559147$ |
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| Real period: | $\Omega$ | ≈ | $0.24693939071111447617680450380$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 256 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $12.416857654917666754030359812 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.416857655 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.246939 \cdot 3.142689 \cdot 256}{4^2} \\ & \approx 12.416857655\end{aligned}$$
Modular invariants
Modular form 405042.2.a.bt
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5308416 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $11$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $17$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $19$ | $4$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 8.12.0.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2584 = 2^{3} \cdot 17 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 989 & 684 \\ 1368 & 419 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 723 & 1634 \\ 1558 & 951 \end{array}\right),\left(\begin{array}{rr} 2581 & 4 \\ 2580 & 5 \end{array}\right),\left(\begin{array}{rr} 1293 & 684 \\ 1634 & 1369 \end{array}\right),\left(\begin{array}{rr} 815 & 0 \\ 0 & 2583 \end{array}\right)$.
The torsion field $K:=\Q(E[2584])$ is a degree-$308631306240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2584\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 361 = 19^{2} \) |
| $3$ | split multiplicative | $4$ | \( 135014 = 2 \cdot 11 \cdot 17 \cdot 19^{2} \) |
| $11$ | split multiplicative | $12$ | \( 36822 = 2 \cdot 3 \cdot 17 \cdot 19^{2} \) |
| $17$ | split multiplicative | $18$ | \( 23826 = 2 \cdot 3 \cdot 11 \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 405042.bt
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1122.h2, its twist by $-19$.
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.