Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-18028595x+29456436717\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-18028595xz^2+29456436717z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-23365059795x+1374669987361902\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2449, -1396)$ | $1.5187366953326059976086055622$ | $\infty$ |
| $(9551/4, 33667/8)$ | $3.0476034292020643562218516582$ | $\infty$ |
| $(9803/4, -9803/8)$ | $0$ | $2$ |
Integral points
\( \left(1557, 71079\right) \), \( \left(1557, -72636\right) \), \( \left(2449, -1053\right) \), \( \left(2449, -1396\right) \), \( \left(2583, 10128\right) \), \( \left(2583, -12711\right) \), \( \left(3443, 88904\right) \), \( \left(3443, -92347\right) \), \( \left(4703, 218369\right) \), \( \left(4703, -223072\right) \)
Invariants
| Conductor: | $N$ | = | \( 462462 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $13444481835937152$ | = | $2^{7} \cdot 3^{4} \cdot 7^{8} \cdot 11^{3} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{4138501360819965875}{85857408} \) | = | $2^{-7} \cdot 3^{-4} \cdot 5^{3} \cdot 7^{-2} \cdot 13^{-2} \cdot 23^{6} \cdot 607^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6243830157101327126027824054$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0519541229828834240346201392$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0693068572444249$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.732788959388018$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.6157159617706621477957399933$ |
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| Real period: | $\Omega$ | ≈ | $0.28684070792731741191527578729$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $10.591801872525723857096536086 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.591801873 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.286841 \cdot 4.615716 \cdot 32}{2^2} \\ & \approx 10.591801873\end{aligned}$$
Modular invariants
Modular form 462462.2.a.bb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 16515072 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $11$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 \( 1144 = 2^{3} \cdot 11 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 316 & 1 \\ 207 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 353 & 4 \\ 706 & 9 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 571 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1141 & 4 \\ 1140 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 716 & 433 \\ 1001 & 144 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[1144])$ is a degree-$44281036800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1144\Z)$.
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 | $4$ | \( 539 = 7^{2} \cdot 11 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 154154 = 2 \cdot 7^{2} \cdot 11^{2} \cdot 13 \) |
| $7$ | additive | $32$ | \( 4719 = 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $42$ | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $13$ | 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 $d=$
2.
Its isogeny class 462462bb
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 66066z2, its twist by $-7$.
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.