Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+6249x+13005341\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+6249xz^2+13005341z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+99981x+832441806\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-137/4, 28703/8)$ | $2.5257145391939936821367251040$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 466578 \) | = | $2 \cdot 3^{2} \cdot 7^{2} \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $-73101173635155456$ | = | $-1 \cdot 2^{9} \cdot 3^{9} \cdot 7^{2} \cdot 23^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{189}{512} \) | = | $2^{-9} \cdot 3^{3} \cdot 7$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9148324744627239880207032697$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.80119220817881867677999919780$ |
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| $abc$ quality: | $Q$ | ≈ | $1.3265882735428125$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.5458888477994717$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.5257145391939936821367251040$ |
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| Real period: | $\Omega$ | ≈ | $0.27112183577315388468525898550$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.7391054500208839826527626348 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.739105450 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.271122 \cdot 2.525715 \cdot 4}{1^2} \\ & \approx 2.739105450\end{aligned}$$
Modular invariants
Modular form 466578.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5132160 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $7$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $23$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $3$ | 3B | 9.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 11592 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 23 \), index $144$, genus $2$, and generators
$\left(\begin{array}{rr} 5843 & 4554 \\ 10350 & 2255 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 12 & 217 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 8074 & 8073 \\ 9315 & 3520 \end{array}\right),\left(\begin{array}{rr} 13 & 12 \\ 11528 & 11533 \end{array}\right),\left(\begin{array}{rr} 1007 & 0 \\ 0 & 11591 \end{array}\right),\left(\begin{array}{rr} 11575 & 18 \\ 11574 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 8302 & 621 \\ 207 & 4622 \end{array}\right),\left(\begin{array}{rr} 8695 & 4554 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[11592])$ is a degree-$22337262452736$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11592\Z)$.
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 | $4$ | \( 77763 = 3 \cdot 7^{2} \cdot 23^{2} \) |
| $3$ | additive | $2$ | \( 25921 = 7^{2} \cdot 23^{2} \) |
| $7$ | additive | $14$ | \( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 466578h
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 882b2, its twist by $-23$.
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.