Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-92607792x+259631758592\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-92607792xz^2+259631758592z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1481724675x+16614950825214\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{12547}{4}, -\frac{12547}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([25094:-12547:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(12546, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 410958 \) | = | $2 \cdot 3^{2} \cdot 17^{2} \cdot 79$ |
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| Minimal Discriminant: | $\Delta$ | = | $21714943867941962500743168$ | = | $2^{13} \cdot 3^{6} \cdot 17^{12} \cdot 79^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{4991662883279390625}{1234063918112768} \) | = | $2^{-13} \cdot 3^{3} \cdot 5^{6} \cdot 17^{-6} \cdot 79^{-2} \cdot 22787^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.5720453661198800140009025086$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6061325497577171281785125812$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1185079752863947$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.155808811680358$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.063738105954961602248391209295$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.25495242381984640899356483718 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.254952424 \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 0.063738 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 0.254952424\end{aligned}$$
Modular invariants
Modular form 410958.2.a.m
For more coefficients, see the Downloads section to the right.
| Modular degree: | 80510976 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{13}$ | nonsplit multiplicative | 1 | 1 | 13 | 13 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $17$ | $4$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
| $79$ | $2$ | $I_{2}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.6.0.6 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10744 = 2^{3} \cdot 17 \cdot 79 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 7585 & 4 \\ 4426 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5849 & 4 \\ 954 & 9 \end{array}\right),\left(\begin{array}{rr} 1345 & 9402 \\ 9400 & 1343 \end{array}\right),\left(\begin{array}{rr} 10741 & 4 \\ 10740 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 5371 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[10744])$ is a degree-$385547281367040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10744\Z)$.
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 | $4$ | \( 2601 = 3^{2} \cdot 17^{2} \) |
| $3$ | additive | $6$ | \( 45662 = 2 \cdot 17^{2} \cdot 79 \) |
| $13$ | good | $2$ | \( 205479 = 3^{2} \cdot 17^{2} \cdot 79 \) |
| $17$ | additive | $162$ | \( 1422 = 2 \cdot 3^{2} \cdot 79 \) |
| $79$ | nonsplit multiplicative | $80$ | \( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 410958.m
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 2686.a1, its twist by $-51$.
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 $0$.