Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+30129x-3332824\)
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(homogenize, simplify) |
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\(y^2z=x^3+30129xz^2-3332824z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+30129x-3332824\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(88, 0)$ | $0$ | $2$ |
Integral points
\( \left(88, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 487872 \) | = | $2^{6} \cdot 3^{2} \cdot 7 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $-6548920421617728$ | = | $-1 \cdot 2^{6} \cdot 3^{7} \cdot 7^{4} \cdot 11^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{36594368}{79233} \) | = | $2^{6} \cdot 3^{-1} \cdot 7^{-4} \cdot 11^{-1} \cdot 83^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7190675431327573332252275770$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.37575982788045543921198289117$ |
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| $abc$ quality: | $Q$ | ≈ | $0.894198818057269$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.325885175366126$ | |||
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: | $\Omega$ | ≈ | $0.21934776339550054446710921021$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.7547821071640043557368736817 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.754782107 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.219348 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 1.754782107\end{aligned}$$
Modular invariants
Modular form 487872.2.a.dn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2457600 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | -1 | 6 | 6 | 0 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $2$ | $I_{1}^{*}$ | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1841 & 8 \\ 1840 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1842 & 1843 \end{array}\right),\left(\begin{array}{rr} 160 & 1845 \\ 331 & 1846 \end{array}\right),\left(\begin{array}{rr} 232 & 1163 \\ 235 & 264 \end{array}\right),\left(\begin{array}{rr} 685 & 690 \\ 1150 & 229 \end{array}\right),\left(\begin{array}{rr} 1585 & 8 \\ 796 & 33 \end{array}\right),\left(\begin{array}{rr} 1228 & 1847 \\ 593 & 1842 \end{array}\right)$.
The torsion field $K:=\Q(E[1848])$ is a degree-$40874803200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1848\Z)$.
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 | $2$ | \( 1089 = 3^{2} \cdot 11^{2} \) |
| $3$ | additive | $8$ | \( 54208 = 2^{6} \cdot 7 \cdot 11^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 69696 = 2^{6} \cdot 3^{2} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 487872.dn
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 7392.a4, its twist by $-264$.
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 $0$.