Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-2219480x+1166199104\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-2219480xz^2+1166199104z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2876446755x+54453332094174\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{4219}{4}, -\frac{4219}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([8438:-4219:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(37986, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 453934 \) | = | $2 \cdot 13^{2} \cdot 17 \cdot 79$ |
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| Minimal Discriminant: | $\Delta$ | = | $111274659511750412288$ | = | $2^{11} \cdot 13^{6} \cdot 17^{2} \cdot 79^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{250505699702316625}{23053462341632} \) | = | $2^{-11} \cdot 5^{3} \cdot 7^{6} \cdot 17^{-2} \cdot 31^{3} \cdot 79^{-4} \cdot 83^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5854656508020707505794123978$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3029909720713023825526686770$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9929394850641525$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.257116624960723$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.18258493919959460896230945100$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.9213590271935137433969512160 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.921359027 \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{4 \cdot 0.182585 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 2.921359027\end{aligned}$$
Modular invariants
Modular form 453934.2.a.m
For more coefficients, see the Downloads section to the right.
| Modular degree: | 20275200 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{11}$ | nonsplit multiplicative | 1 | 1 | 11 | 11 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $17$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $79$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.6.0.6 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 136 = 2^{3} \cdot 17 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 133 & 4 \\ 132 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 67 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 121 & 18 \\ 16 & 119 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 105 & 4 \\ 74 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[136])$ is a degree-$10027008$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/136\Z)$.
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 | $4$ | \( 169 = 13^{2} \) |
| $11$ | good | $2$ | \( 226967 = 13^{2} \cdot 17 \cdot 79 \) |
| $13$ | additive | $86$ | \( 2686 = 2 \cdot 17 \cdot 79 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 26702 = 2 \cdot 13^{2} \cdot 79 \) |
| $79$ | split multiplicative | $80$ | \( 5746 = 2 \cdot 13^{2} \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 453934.m
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 2686.d1, its twist by $13$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.