Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+33614080153x+25447382790781\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+33614080153xz^2+25447382790781z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+537825282445x+1629170323892430\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(114724815098185626496439304236774817250318788050480258010159793686923622447/429113906469797683537470271707130020998339921369576370746796635613604, 1489481597479431929616108990905929332175864889335627570836660104046460434444372166657030861553613404114733719741/8889122345975452764791158472083172706130525839327867891756273429074503376814425406456821760437850479192)$ | $167.28893213142774203645029850$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 477950 \) | = | $2 \cdot 5^{2} \cdot 11^{2} \cdot 79$ |
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| Discriminant: | $\Delta$ | = | $-2431048816291367950435318811852800$ | = | $-1 \cdot 2^{31} \cdot 5^{2} \cdot 11^{9} \cdot 79^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{71254653245666829920041965}{41240080525456710828032} \) | = | $2^{-31} \cdot 3^{3} \cdot 5 \cdot 79^{-7} \cdot 80815219^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $5.0956023802927341035507980472$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.0289412736216061330708804749$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1712097808122228$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.44846294921661$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $167.28893213142774203645029850$ |
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| Real period: | $\Omega$ | ≈ | $0.0086723691616873681522468820450$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.9015827522164100662146168968 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.901582752 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.008672 \cdot 167.288932 \cdot 2}{1^2} \\ & \approx 2.901582752\end{aligned}$$
Modular invariants
Modular form 477950.2.a.t
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1706265792 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{31}$ | nonsplit multiplicative | 1 | 1 | 31 | 31 |
| $5$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
| $11$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $79$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6952 = 2^{3} \cdot 11 \cdot 79 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 5215 & 2 \\ 5215 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 5689 & 2 \\ 5689 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6951 & 2 \\ 6950 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 6951 & 0 \end{array}\right),\left(\begin{array}{rr} 3477 & 2 \\ 3477 & 3 \end{array}\right),\left(\begin{array}{rr} 793 & 2 \\ 793 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[6952])$ is a degree-$389799641088000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6952\Z)$.
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 | $4$ | \( 21725 = 5^{2} \cdot 11 \cdot 79 \) |
| $5$ | additive | $10$ | \( 19118 = 2 \cdot 11^{2} \cdot 79 \) |
| $7$ | good | $2$ | \( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \) |
| $11$ | additive | $42$ | \( 3950 = 2 \cdot 5^{2} \cdot 79 \) |
| $31$ | good | $2$ | \( 238975 = 5^{2} \cdot 11^{2} \cdot 79 \) |
| $79$ | nonsplit multiplicative | $80$ | \( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 477950.t consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 477950.cu1, its twist by $-11$.
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.