Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+370258x-84074316\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+370258xz^2-84074316z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+479853693x-3929769096066\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(325, 8247)$ | $4.2785550290349802405899140854$ | $\infty$ |
| $(4861/9, 439501/27)$ | $5.7021365448154724440446545391$ | $\infty$ |
| $(204, -102)$ | $0$ | $2$ |
Integral points
\( \left(204, -102\right) \), \( \left(325, 8247\right) \), \( \left(325, -8572\right) \), \( \left(565, 17207\right) \), \( \left(565, -17772\right) \), \( \left(4300, 282522\right) \), \( \left(4300, -286822\right) \)
Invariants
| Conductor: | $N$ | = | \( 431970 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-6313369222030295040$ | = | $-1 \cdot 2^{24} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11^{6} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{3168685387909439}{3563732336640} \) | = | $2^{-24} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-2} \cdot 17^{-2} \cdot 191^{3} \cdot 769^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2938857826493179299091729040$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0949381462501326578782011150$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9594119735978286$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.859358416946873$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $20.283741869315231555828221512$ |
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| Real period: | $\Omega$ | ≈ | $0.12841656904885864292606226428$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot1\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $10.419074153320578068149135068 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.419074153 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.128417 \cdot 20.283742 \cdot 16}{2^2} \\ & \approx 10.419074153\end{aligned}$$
Modular invariants
Modular form 431970.2.a.y
For more coefficients, see the Downloads section to the right.
| Modular degree: | 11796480 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 4 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 6 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{24}$ | nonsplit multiplicative | 1 | 1 | 24 | 24 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $17$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 | 16.24.0.13 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 44880 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 20416 \\ 11220 & 11221 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 35201 & 20416 \\ 36410 & 5391 \end{array}\right),\left(\begin{array}{rr} 35773 & 20416 \\ 40304 & 11925 \end{array}\right),\left(\begin{array}{rr} 4079 & 0 \\ 0 & 44879 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 44782 & 44867 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 44876 & 44877 \end{array}\right),\left(\begin{array}{rr} 25312 & 20405 \\ 40755 & 28546 \end{array}\right),\left(\begin{array}{rr} 1376 & 20405 \\ 40755 & 28546 \end{array}\right),\left(\begin{array}{rr} 44865 & 16 \\ 44864 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[44880])$ is a degree-$3049493889024000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/44880\Z)$.
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 | $4$ | \( 1815 = 3 \cdot 5 \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 71995 = 5 \cdot 7 \cdot 11^{2} \cdot 17 \) |
| $5$ | split multiplicative | $6$ | \( 86394 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \cdot 17 \) |
| $7$ | split multiplicative | $8$ | \( 61710 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 17 \) |
| $11$ | additive | $62$ | \( 3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 25410 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 431970.y
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 3570.t5, 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.