Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-203431x-35396332\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-203431xz^2-35396332z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-263646603x-1647496558458\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(549, 4057)$ | $3.7523648011687734391704492572$ | $\infty$ |
| $(-1053/4, 1049/8)$ | $0$ | $2$ |
Integral points
\( \left(549, 4057\right) \), \( \left(549, -4607\right) \)
Invariants
| Conductor: | $N$ | = | \( 492765 \) | = | $3 \cdot 5 \cdot 7 \cdot 13 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $141086127760305$ | = | $3 \cdot 5 \cdot 7 \cdot 13^{4} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{19790357598649}{2998905} \) | = | $3^{-1} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{3} \cdot 13^{-4} \cdot 2459^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7277181469572196833918347064$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.25549865737399945338732099046$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9159555310901241$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6835239516016247$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.7523648011687734391704492572$ |
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| Real period: | $\Omega$ | ≈ | $0.22480150222784989563825319818$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot1\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.3741489768385902118647294701 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $4$ = $2^2$ (rounded) |
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BSD formula
$$\begin{aligned} 3.374148977 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.224802 \cdot 3.752365 \cdot 4}{2^2} \\ & \approx 3.374148977\end{aligned}$$
Modular invariants
Modular form 492765.2.a.e
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2654208 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $19$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 207480 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 200204 & 43681 \\ 105583 & 10926 \end{array}\right),\left(\begin{array}{rr} 43679 & 0 \\ 0 & 207479 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 207474 & 207475 \end{array}\right),\left(\begin{array}{rr} 47881 & 141968 \\ 158764 & 152913 \end{array}\right),\left(\begin{array}{rr} 156979 & 156978 \\ 23218 & 159715 \end{array}\right),\left(\begin{array}{rr} 127928 & 131043 \\ 159125 & 43682 \end{array}\right),\left(\begin{array}{rr} 207473 & 8 \\ 207472 & 9 \end{array}\right),\left(\begin{array}{rr} 15296 & 131043 \\ 10925 & 43682 \end{array}\right),\left(\begin{array}{rr} 20483 & 173356 \\ 107882 & 36861 \end{array}\right)$.
The torsion field $K:=\Q(E[207480])$ is a degree-$4796069498899660800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/207480\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 37905 = 3 \cdot 5 \cdot 7 \cdot 19^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 164255 = 5 \cdot 7 \cdot 13 \cdot 19^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 98553 = 3 \cdot 7 \cdot 13 \cdot 19^{2} \) |
| $7$ | split multiplicative | $8$ | \( 70395 = 3 \cdot 5 \cdot 13 \cdot 19^{2} \) |
| $13$ | nonsplit multiplicative | $14$ | \( 37905 = 3 \cdot 5 \cdot 7 \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 1365 = 3 \cdot 5 \cdot 7 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 492765e
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1365e4, its twist by $-19$.
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.