Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-20139627099x+1100058556561418\)
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(homogenize, simplify) |
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\(y^2z=x^3-20139627099xz^2+1100058556561418z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-20139627099x+1100058556561418\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4326114833561395286331809996503529372286261479/629533720994842783663832461395343002025, 284482249725170774083595938234610675820040087378184014470735481195058/15795322484952051905405818979083085498388782958739347408875)$ | $101.55082162832109612154139012$ | $\infty$ |
| $(82246, 0)$ | $0$ | $2$ |
Integral points
\( \left(82246, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 457776 \) | = | $2^{4} \cdot 3^{2} \cdot 11 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $22745373206149456942399930368$ | = | $2^{14} \cdot 3^{26} \cdot 11^{3} \cdot 17^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{12534210458299016895673}{315581882565708} \) | = | $2^{-2} \cdot 3^{-20} \cdot 11^{-3} \cdot 17^{-1} \cdot 23229097^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.5493259860627980151456652050$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.8902659891406898199060431561$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0294913049284855$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.3518957724474365$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $101.55082162832109612154139012$ |
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| Real period: | $\Omega$ | ≈ | $0.035292164314550546999151435406$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2^{2}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $14.335793132737286641531446872 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.335793133 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.035292 \cdot 101.550822 \cdot 16}{2^2} \\ & \approx 14.335793133\end{aligned}$$
Modular invariants
Modular form 457776.2.a.fg
For more coefficients, see the Downloads section to the right.
| Modular degree: | 849346560 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}^{*}$ | additive | -1 | 4 | 14 | 2 |
| $3$ | $4$ | $I_{20}^{*}$ | additive | -1 | 2 | 26 | 20 |
| $11$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $17$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4488 = 2^{3} \cdot 3 \cdot 11 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 4482 & 4483 \end{array}\right),\left(\begin{array}{rr} 260 & 4487 \\ 2617 & 4482 \end{array}\right),\left(\begin{array}{rr} 3272 & 3 \\ 2045 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1495 & 4480 \\ 1492 & 4455 \end{array}\right),\left(\begin{array}{rr} 560 & 2797 \\ 557 & 528 \end{array}\right),\left(\begin{array}{rr} 4481 & 8 \\ 4480 & 9 \end{array}\right),\left(\begin{array}{rr} 3931 & 3930 \\ 2818 & 571 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[4488])$ is a degree-$1588278067200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4488\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 28611 = 3^{2} \cdot 11 \cdot 17^{2} \) |
| $3$ | additive | $8$ | \( 4624 = 2^{4} \cdot 17^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 41616 = 2^{4} \cdot 3^{2} \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 457776.fg
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1122.c1, its twist by $204$.
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.