Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+4233183x-37875637323\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+4233183xz^2-37875637323z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+5486204493x-1767208028012658\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(173469261383/14691889, 72185885143720835/56314010537)$ | $20.699323282942088009646758646$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 462462 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-624644235015830781687936$ | = | $-1 \cdot 2^{7} \cdot 3^{7} \cdot 7^{13} \cdot 11^{6} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{40251338884511}{2997011332224} \) | = | $2^{-7} \cdot 3^{-7} \cdot 7^{-7} \cdot 13^{-1} \cdot 43^{3} \cdot 797^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2454953104259016971559656527$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0735925994990597725723174920$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0387829692161805$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.771301846407381$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $20.699323282942088009646758646$ |
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| Real period: | $\Omega$ | ≈ | $0.043493343125385586895376310453$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot1\cdot2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.6011310800331323244343000069 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.601131080 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.043493 \cdot 20.699323 \cdot 4}{1^2} \\ & \approx 3.601131080\end{aligned}$$
Modular invariants
Modular form 462462.2.a.bi
For more coefficients, see the Downloads section to the right.
| Modular degree: | 67173120 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $3$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $7$ | $4$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $11$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 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 |
|---|---|---|
| $7$ | 7B.6.1 | 7.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 24024 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 13 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 848 & 17479 \\ 12089 & 6546 \end{array}\right),\left(\begin{array}{rr} 6544 & 539 \\ 23485 & 2034 \end{array}\right),\left(\begin{array}{rr} 17480 & 17479 \\ 12551 & 6546 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19655 & 0 \\ 0 & 24023 \end{array}\right),\left(\begin{array}{rr} 24011 & 14 \\ 24010 & 15 \end{array}\right),\left(\begin{array}{rr} 17480 & 17479 \\ 18557 & 6546 \end{array}\right),\left(\begin{array}{rr} 9472 & 17479 \\ 22561 & 6546 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right)$.
The torsion field $K:=\Q(E[24024])$ is a degree-$535623421132800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24024\Z)$.
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 | $4$ | \( 231231 = 3 \cdot 7^{2} \cdot 11^{2} \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 154154 = 2 \cdot 7^{2} \cdot 11^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 1573 = 11^{2} \cdot 13 \) |
| $11$ | additive | $62$ | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 35574 = 2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 462462.bi
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 546.f2, its twist by $77$.
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.