Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-733734x-57479709\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-733734xz^2-57479709z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-950919291x-2667509505450\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-815, 407)$ | $0$ | $2$ |
| $(-79, 39)$ | $0$ | $2$ |
Integral points
\( \left(-815, 407\right) \), \( \left(-79, 39\right) \)
Invariants
| Conductor: | $N$ | = | \( 453882 \) | = | $2 \cdot 3 \cdot 11 \cdot 13 \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $23868949078177879104$ | = | $2^{6} \cdot 3^{6} \cdot 11^{2} \cdot 13^{4} \cdot 23^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{295102348042033}{161237583936} \) | = | $2^{-6} \cdot 3^{-6} \cdot 7^{3} \cdot 11^{-2} \cdot 13^{-4} \cdot 9511^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4090323223698571424496137022$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.84128521440528229704623728630$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0588014549250642$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.0022216666714625$ | |||
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: | $\Omega$ | ≈ | $0.17419365235190175694968904506$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ ( 2 \cdot 3 )\cdot2\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.0903238282228210833962685408 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.090323828 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.174194 \cdot 1.000000 \cdot 192}{4^2} \\ & \approx 2.090323828\end{aligned}$$
Modular invariants
Modular form 453882.2.a.bx
For more coefficients, see the Downloads section to the right.
| Modular degree: | 14192640 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $11$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $23$ | $4$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 8.12.0.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6072 = 2^{3} \cdot 3 \cdot 11 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3313 & 2116 \\ 1610 & 4233 \end{array}\right),\left(\begin{array}{rr} 5613 & 1058 \\ 1840 & 2899 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4095 & 1058 \\ 5014 & 5015 \end{array}\right),\left(\begin{array}{rr} 2111 & 0 \\ 0 & 6071 \end{array}\right),\left(\begin{array}{rr} 6069 & 4 \\ 6068 & 5 \end{array}\right),\left(\begin{array}{rr} 4049 & 2116 \\ 3082 & 4233 \end{array}\right)$.
The torsion field $K:=\Q(E[6072])$ is a degree-$5416884633600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6072\Z)$.
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 | $4$ | \( 529 = 23^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 75647 = 11 \cdot 13 \cdot 23^{2} \) |
| $11$ | split multiplicative | $12$ | \( 41262 = 2 \cdot 3 \cdot 13 \cdot 23^{2} \) |
| $13$ | nonsplit multiplicative | $14$ | \( 34914 = 2 \cdot 3 \cdot 11 \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 858 = 2 \cdot 3 \cdot 11 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 453882.bx
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 858.h2, its twist by $-23$.
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 $0$.