Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-3456300x+2468306896\)
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(homogenize, simplify) |
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\(y^2z=x^3-3456300xz^2+2468306896z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3456300x+2468306896\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{3494}{9}, \frac{1653760}{27}\right) \) | $4.3325312626498353160470304113$ | $\infty$ |
| \( \left(1034, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-10482:1653760:27]\) | $4.3325312626498353160470304113$ | $\infty$ |
| \([1034:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{3494}{9}, \frac{1653760}{27}\right) \) | $4.3325312626498353160470304113$ | $\infty$ |
| \( \left(1034, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(1034, 0\right) \)
\([1034:0:1]\)
\( \left(1034, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 470592 \) | = | $2^{6} \cdot 3^{2} \cdot 19 \cdot 43$ |
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| Minimal Discriminant: | $\Delta$ | = | $10518751671629119488$ | = | $2^{26} \cdot 3^{12} \cdot 19^{3} \cdot 43 $ |
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| j-invariant: | $j$ | = | \( \frac{23894093340015625}{55042322688} \) | = | $2^{-8} \cdot 3^{-6} \cdot 5^{6} \cdot 19^{-3} \cdot 41^{3} \cdot 43^{-1} \cdot 281^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5309718258364192774899222252$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.94194491066244646766645142455$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9843247587061957$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.347100868469556$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.3325312626498353160470304113$ |
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| Real period: | $\Omega$ | ≈ | $0.22875941897449913135777664787$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2^{2}\cdot2^{2}\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.893288011991553000729286267 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.893288012 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.228759 \cdot 4.332531 \cdot 48}{2^2} \\ & \approx 11.893288012\end{aligned}$$
Modular invariants
Modular form 470592.2.a.dq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 10616832 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{16}^{*}$ | additive | -1 | 6 | 26 | 8 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $19$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $43$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 19608 = 2^{3} \cdot 3 \cdot 19 \cdot 43 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19597 & 12 \\ 19596 & 13 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 19558 & 19599 \end{array}\right),\left(\begin{array}{rr} 8266 & 3 \\ 5133 & 9796 \end{array}\right),\left(\begin{array}{rr} 15513 & 18788 \\ 15550 & 18799 \end{array}\right),\left(\begin{array}{rr} 9815 & 19596 \\ 9816 & 19595 \end{array}\right),\left(\begin{array}{rr} 9803 & 0 \\ 0 & 19607 \end{array}\right),\left(\begin{array}{rr} 19597 & 19606 \\ 9854 & 9813 \end{array}\right),\left(\begin{array}{rr} 13063 & 19606 \\ 13068 & 19607 \end{array}\right),\left(\begin{array}{rr} 13214 & 19605 \\ 15303 & 9812 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[19608])$ is a degree-$315580049326080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/19608\Z)$.
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 | $2$ | \( 7353 = 3^{2} \cdot 19 \cdot 43 \) |
| $3$ | additive | $2$ | \( 2752 = 2^{6} \cdot 43 \) |
| $19$ | split multiplicative | $20$ | \( 24768 = 2^{6} \cdot 3^{2} \cdot 43 \) |
| $43$ | split multiplicative | $44$ | \( 10944 = 2^{6} \cdot 3^{2} \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 470592dq
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 4902h1, its twist by $24$.
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.