Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-10402346x-10853055360\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-10402346xz^2-10853055360z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-13481441091x-506157929263170\)
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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 |
|---|---|---|
| $(-1385, 30690)$ | $5.2527754153021543865352792117$ | $\infty$ |
| $(-1216, 608)$ | $0$ | $2$ |
| $(3656, -1828)$ | $0$ | $2$ |
Integral points
\( \left(-1385, 30690\right) \), \( \left(-1385, -29305\right) \), \( \left(-1216, 608\right) \), \( \left(3656, -1828\right) \), \( \left(34106, 6252602\right) \), \( \left(34106, -6286708\right) \)
Invariants
| Conductor: | $N$ | = | \( 459186 \) | = | $2 \cdot 3 \cdot 7 \cdot 13 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $21195905228066339902404$ | = | $2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{4} \cdot 29^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{209280938961918097}{35633951258724} \) | = | $2^{-2} \cdot 3^{-2} \cdot 7^{-2} \cdot 13^{-4} \cdot 29^{-4} \cdot 61^{3} \cdot 9733^{3}$ |
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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: | $Q$ | ≈ | $0.9333363087644262$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.6088255078966744$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.2527754153021543865352792117$ |
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| Real period: | $\Omega$ | ≈ | $0.085040960896139953563824667895$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2\cdot2\cdot2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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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
$$\begin{aligned} 3.573608550 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.085041 \cdot 5.252775 \cdot 128}{4^2} \\ & \approx 3.573608550\end{aligned}$$
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 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $29$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 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 |
|---|---|---|
| $2$ | 2Cs | 8.24.0.14 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 63336 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \cdot 29 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 15842 \\ 0 & 47503 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 47510 \\ 47494 & 63315 \end{array}\right),\left(\begin{array}{rr} 21119 & 2 \\ 63318 & 63331 \end{array}\right),\left(\begin{array}{rr} 63329 & 8 \\ 63328 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 2 \\ 45222 & 63331 \end{array}\right),\left(\begin{array}{rr} 19489 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 63332 & 63333 \end{array}\right),\left(\begin{array}{rr} 34943 & 63328 \\ 13100 & 63303 \end{array}\right)$.
The torsion field $K:=\Q(E[63336])$ is a degree-$13838561480540160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/63336\Z)$.
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 | $4$ | \( 841 = 29^{2} \) |
| $3$ | 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 and 4.
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 15834v3, 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.