Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-384492x+123225424\)
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(homogenize, simplify) |
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\(y^2z=x^3-384492xz^2+123225424z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-384492x+123225424\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-742, 0)$ | $0$ | $2$ |
Integral points
\( \left(-742, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 411840 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-2921880417730560000$ | = | $-1 \cdot 2^{24} \cdot 3^{11} \cdot 5^{4} \cdot 11^{2} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{32894113444921}{15289560000} \) | = | $-1 \cdot 2^{-6} \cdot 3^{-5} \cdot 5^{-4} \cdot 11^{-2} \cdot 13^{-1} \cdot 179^{6}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2494473015265734941213051890$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.66042038635260068429783438835$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0783963817005138$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.92795983425011$ | |||
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: | $\Omega$ | ≈ | $0.23720964863285745435783337066$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.7953543781257192697253339305 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.795354378 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.237210 \cdot 1.000000 \cdot 64}{2^2} \\ & \approx 3.795354378\end{aligned}$$
Modular invariants
Modular form 411840.2.a.jv
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5898240 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{14}^{*}$ | additive | -1 | 6 | 24 | 6 |
| $3$ | $2$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $1$ | $I_{1}$ | nonsplit 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1717 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2290 & 1 \\ 2287 & 0 \end{array}\right),\left(\begin{array}{rr} 433 & 3004 \\ 2144 & 1287 \end{array}\right),\left(\begin{array}{rr} 937 & 4 \\ 1874 & 9 \end{array}\right),\left(\begin{array}{rr} 1850 & 1 \\ 791 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3429 & 4 \\ 3428 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[3432])$ is a degree-$2125489766400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3432\Z)$.
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 | $2$ | \( 117 = 3^{2} \cdot 13 \) |
| $3$ | additive | $8$ | \( 45760 = 2^{6} \cdot 5 \cdot 11 \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 82368 = 2^{6} \cdot 3^{2} \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 37440 = 2^{6} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 31680 = 2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 411840.jv
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 4290.o2, its twist by $24$.
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 $0$.