Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-1909x+13211\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-1909xz^2+13211z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2474091x+653492070\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(7, 12)$ | $1.7104114244755921185326193045$ | $\infty$ |
| $(55, 252)$ | $2.3889525993491850030350659952$ | $\infty$ |
| $(39, -20)$ | $0$ | $2$ |
Integral points
\( \left(-47, 66\right) \), \( \left(-47, -20\right) \), \( \left(-33, 220\right) \), \( \left(-33, -188\right) \), \( \left(7, 12\right) \), \( \left(7, -20\right) \), \( \left(39, -20\right) \), \( \left(55, 252\right) \), \( \left(55, -308\right) \), \( \left(211, 2904\right) \), \( \left(211, -3116\right) \), \( \left(2955, 159172\right) \), \( \left(2955, -162128\right) \)
Invariants
| Conductor: | $N$ | = | \( 454854 \) | = | $2 \cdot 3 \cdot 41 \cdot 43^{2}$ |
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| Discriminant: | $\Delta$ | = | $360506363904$ | = | $2^{12} \cdot 3^{3} \cdot 41 \cdot 43^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{9677214091}{4534272} \) | = | $2^{-12} \cdot 3^{-3} \cdot 41^{-1} \cdot 2131^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.91190526861689113886218737569$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.028394760306499467006023252646$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8940677535292496$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.6310520400305344$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.5006416734992928448559467679$ |
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| Real period: | $\Omega$ | ≈ | $0.85419021956192715837187936962$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ ( 2^{2} \cdot 3 )\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $17.941283278163958477361632932 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 17.941283278 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.854190 \cdot 3.500642 \cdot 24}{2^2} \\ & \approx 17.941283278\end{aligned}$$
Modular invariants
Modular form 454854.2.a.n
For more coefficients, see the Downloads section to the right.
| Modular degree: | 532224 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $41$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $43$ | $2$ | $III$ | additive | 1 | 2 | 3 | 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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 21156 = 2^{2} \cdot 3 \cdot 41 \cdot 43 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 7054 & 1 \\ 7051 & 0 \end{array}\right),\left(\begin{array}{rr} 5293 & 15868 \\ 5288 & 15867 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6892 & 1 \\ 8855 & 0 \end{array}\right),\left(\begin{array}{rr} 21153 & 4 \\ 21152 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 19610 & 1 \\ 11351 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[21156])$ is a degree-$3531051624038400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/21156\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 5289 = 3 \cdot 41 \cdot 43 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 75809 = 41 \cdot 43^{2} \) |
| $41$ | nonsplit multiplicative | $42$ | \( 11094 = 2 \cdot 3 \cdot 43^{2} \) |
| $43$ | additive | $506$ | \( 246 = 2 \cdot 3 \cdot 41 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 454854n
consists of 2 curves linked by isogenies of
degree 2.
Twists
This elliptic curve is its own minimal quadratic twist.
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.