Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-26018876x-1556753940102\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-26018876xz^2-1556753940102z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-33720462675x-72631810667999250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(7723457677/309136, 626762838184061/171879616)$ | $21.435272852627335837286930225$ | $\infty$ |
| $(12337, -6169)$ | $0$ | $2$ |
Integral points
\( \left(12337, -6169\right) \)
Invariants
| Conductor: | $N$ | = | \( 462550 \) | = | $2 \cdot 5^{2} \cdot 11 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1045814350542005862400000000$ | = | $-1 \cdot 2^{24} \cdot 5^{8} \cdot 11 \cdot 29^{9} $ |
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| j-invariant: | $j$ | = | \( -\frac{209595169258201}{112524368281600} \) | = | $-1 \cdot 2^{-24} \cdot 5^{-2} \cdot 11^{-1} \cdot 29^{-3} \cdot 191^{3} \cdot 311^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8636404932488952149235583422$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3752736220386080140315426594$ |
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| $abc$ quality: | $Q$ | ≈ | $0.993673517358364$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.340974172693366$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $21.435272852627335837286930225$ |
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| Real period: | $\Omega$ | ≈ | $0.022113154418737842606040185409$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.7920119887834199356850896229 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.792011989 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.022113 \cdot 21.435273 \cdot 32}{2^2} \\ & \approx 3.792011989\end{aligned}$$
Modular invariants
Modular form 462550.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 209018880 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{24}$ | nonsplit multiplicative | 1 | 1 | 24 | 24 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $29$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
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 \( 19140 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 29 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 19129 & 12 \\ 19128 & 13 \end{array}\right),\left(\begin{array}{rr} 11175 & 17548 \\ 11138 & 17537 \end{array}\right),\left(\begin{array}{rr} 11870 & 19137 \\ 13887 & 8 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 11483 & 19128 \\ 11478 & 19067 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 19090 & 19131 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8710 & 3 \\ 5193 & 19132 \end{array}\right),\left(\begin{array}{rr} 12761 & 12 \\ 12760 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[19140])$ is a degree-$207439626240000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/19140\Z)$.
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 | $4$ | \( 231275 = 5^{2} \cdot 11 \cdot 29^{2} \) |
| $3$ | good | $2$ | \( 231275 = 5^{2} \cdot 11 \cdot 29^{2} \) |
| $5$ | additive | $18$ | \( 18502 = 2 \cdot 11 \cdot 29^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 42050 = 2 \cdot 5^{2} \cdot 29^{2} \) |
| $29$ | additive | $450$ | \( 550 = 2 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 462550d
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 3190c3, its twist by $145$.
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.