Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2+17208x+1969148\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z+17208xz^2+1969148z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+1393821x+1439690382\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-82, 0)$ | $0$ | $2$ |
Integral points
\( \left(-82, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 488072 \) | = | $2^{3} \cdot 13^{2} \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-2010968279636992$ | = | $-1 \cdot 2^{10} \cdot 13^{3} \cdot 19^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{5324}{19} \) | = | $2^{2} \cdot 11^{3} \cdot 19^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6193880373575446899975730377$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0716914420576808152013393065$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7022352548084589$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.2458755233549796$ | |||
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: | $\Omega$ | ≈ | $0.33073783959052665589884633110$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.66147567918105331179769266220 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.661475679 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.330738 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 0.661475679\end{aligned}$$
Modular invariants
Modular form 488072.2.a.be
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2039040 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | -1 | 3 | 10 | 0 |
| $13$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $19$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 \( 1976 = 2^{3} \cdot 13 \cdot 19 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 249 & 1730 \\ 1728 & 247 \end{array}\right),\left(\begin{array}{rr} 1973 & 4 \\ 1972 & 5 \end{array}\right),\left(\begin{array}{rr} 1562 & 1 \\ 207 & 0 \end{array}\right),\left(\begin{array}{rr} 1828 & 1 \\ 1519 & 0 \end{array}\right),\left(\begin{array}{rr} 989 & 4 \\ 2 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[1976])$ is a degree-$413021306880$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1976\Z)$.
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 | $2$ | \( 4693 = 13 \cdot 19^{2} \) |
| $13$ | additive | $50$ | \( 2888 = 2^{3} \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 1352 = 2^{3} \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 488072be
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 25688d1, its twist by $-19$.
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 $0$.