Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+3356414x+8541460936\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+3356414xz^2+8541460936z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+4349911869x+398445152748606\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-926, 68584)$ | $1.3131938538255385966338198029$ | $\infty$ |
| $(-6053/4, 6053/8)$ | $0$ | $2$ |
Integral points
\( \left(-926, 68584\right) \), \( \left(-926, -67658\right) \), \( \left(3743, 269351\right) \), \( \left(3743, -273094\right) \), \( \left(15894, 2011294\right) \), \( \left(15894, -2027188\right) \)
Invariants
| Conductor: | $N$ | = | \( 459186 \) | = | $2 \cdot 3 \cdot 7 \cdot 13 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $-33926861572757544484548$ | = | $-1 \cdot 2^{2} \cdot 3^{8} \cdot 7^{8} \cdot 13 \cdot 29^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{7030109576481263}{57036871916388} \) | = | $2^{-2} \cdot 3^{-8} \cdot 7^{-8} \cdot 13^{-1} \cdot 23^{3} \cdot 29^{-1} \cdot 8329^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0046234437692916987963600823$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3209755287760546852047240661$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9460552634157916$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.545390536636623$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.3131938538255385966338198029$ |
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| Real period: | $\Omega$ | ≈ | $0.085040960896139953563824667895$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2\cdot2\cdot2^{3}\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.5736085495113265267939494787 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.573608550 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.085041 \cdot 1.313194 \cdot 128}{2^2} \\ & \approx 3.573608550\end{aligned}$$
Modular invariants
Modular form 459186.2.a.f
For more coefficients, see the Downloads section to the right.
| Modular degree: | 43008000 |
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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 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 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $7$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $29$ | $4$ | $I_{1}^{*}$ | 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 | 8.24.0.94 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 126672 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \cdot 29 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 58480 & 5 \\ 9699 & 126658 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 31932 & 31993 \end{array}\right),\left(\begin{array}{rr} 15842 & 95011 \\ 110863 & 63358 \end{array}\right),\left(\begin{array}{rr} 126657 & 16 \\ 126656 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 126574 & 126659 \end{array}\right),\left(\begin{array}{rr} 26200 & 126671 \\ 100385 & 126662 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 84449 & 16 \\ 42232 & 129 \end{array}\right),\left(\begin{array}{rr} 36193 & 16 \\ 36200 & 129 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 126668 & 126669 \end{array}\right)$.
The torsion field $K:=\Q(E[126672])$ is a degree-$221416983688642560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/126672\Z)$.
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 | $4$ | \( 10933 = 13 \cdot 29^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 153062 = 2 \cdot 7 \cdot 13 \cdot 29^{2} \) |
| $7$ | split multiplicative | $8$ | \( 65598 = 2 \cdot 3 \cdot 13 \cdot 29^{2} \) |
| $13$ | split multiplicative | $14$ | \( 35322 = 2 \cdot 3 \cdot 7 \cdot 29^{2} \) |
| $29$ | additive | $450$ | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 459186f
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 15834v4, 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.