Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-364731x-181635067\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-364731xz^2-181635067z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5835699x-11630479986\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1397461/1225, 1251689186/42875)$ | $9.4475358009279928317251057780$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 469278 \) | = | $2 \cdot 3^{2} \cdot 29^{2} \cdot 31$ |
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| Discriminant: | $\Delta$ | = | $-11161288645279878816$ | = | $-1 \cdot 2^{5} \cdot 3^{9} \cdot 29^{6} \cdot 31^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{458314011}{953312} \) | = | $-1 \cdot 2^{-5} \cdot 3^{3} \cdot 31^{-3} \cdot 257^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3440344655325160688473004806$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.16357266596180321329076946327$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9685438880978434$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.948186267785942$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $9.4475358009279928317251057780$ |
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| Real period: | $\Omega$ | ≈ | $0.091134760770991424676338835092$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.4439956603717979633567078957 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.443995660 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.091135 \cdot 9.447536 \cdot 4}{1^2} \\ & \approx 3.443995660\end{aligned}$$
Modular invariants
Modular form 469278.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 17418240 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $29$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $31$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 21576 = 2^{3} \cdot 3 \cdot 29 \cdot 31 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 16183 & 13398 \\ 12093 & 18619 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 10789 & 13398 \\ 17487 & 18619 \end{array}\right),\left(\begin{array}{rr} 21571 & 6 \\ 21570 & 7 \end{array}\right),\left(\begin{array}{rr} 15282 & 14471 \\ 7685 & 2058 \end{array}\right),\left(\begin{array}{rr} 2231 & 0 \\ 0 & 21575 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 6265 & 13398 \\ 3915 & 18619 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[21576])$ is a degree-$2806092398592000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/21576\Z)$.
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 | $4$ | \( 78213 = 3 \cdot 29^{2} \cdot 31 \) |
| $3$ | additive | $2$ | \( 1682 = 2 \cdot 29^{2} \) |
| $5$ | good | $2$ | \( 234639 = 3^{2} \cdot 29^{2} \cdot 31 \) |
| $29$ | additive | $422$ | \( 558 = 2 \cdot 3^{2} \cdot 31 \) |
| $31$ | nonsplit multiplicative | $32$ | \( 15138 = 2 \cdot 3^{2} \cdot 29^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 469278.a
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 558.d1, its twist by $-87$.
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.