Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-507901x-138794774\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-507901xz^2-138794774z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-41140008x-101304810243\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4338, 281554)$ | $3.4683390245842196532774655651$ | $\infty$ |
| $(-394, 0)$ | $0$ | $2$ |
Integral points
\( \left(-394, 0\right) \), \((4338,\pm 281554)\)
Invariants
| Conductor: | $N$ | = | \( 496860 \) | = | $2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $42930915454299600$ | = | $2^{4} \cdot 3^{3} \cdot 5^{2} \cdot 7^{7} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{1594753024}{4725} \) | = | $2^{17} \cdot 3^{-3} \cdot 5^{-2} \cdot 7^{-1} \cdot 23^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0613253499834287916607474004$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.42515346346164466539108339926$ |
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| $abc$ quality: | $Q$ | ≈ | $1.002089692021393$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.890475994254413$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.4683390245842196532774655651$ |
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| Real period: | $\Omega$ | ≈ | $0.17886775922848865408639869035$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot1\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.4814961182884055671523139138 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.481496118 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.178868 \cdot 3.468339 \cdot 16}{2^2} \\ & \approx 2.481496118\end{aligned}$$
Modular invariants
Modular form 496860.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7464960 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5460 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 5449 & 12 \\ 5448 & 13 \end{array}\right),\left(\begin{array}{rr} 1259 & 0 \\ 0 & 5459 \end{array}\right),\left(\begin{array}{rr} 4239 & 598 \\ 2366 & 2081 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 2666 & 2951 \\ 4329 & 820 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3277 & 4212 \\ 5382 & 3433 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 5410 & 5451 \end{array}\right),\left(\begin{array}{rr} 3470 & 1677 \\ 4407 & 428 \end{array}\right)$.
The torsion field $K:=\Q(E[5460])$ is a degree-$1217325957120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5460\Z)$.
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 | $2$ | \( 24843 = 3 \cdot 7^{2} \cdot 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 165620 = 2^{2} \cdot 5 \cdot 7^{2} \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 99372 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13^{2} \) |
| $7$ | additive | $32$ | \( 10140 = 2^{2} \cdot 3 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 496860a
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 420c1, its twist by $-91$.
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.