Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+110860x+23322340\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+110860xz^2+23322340z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+143675181x+1087696081134\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(118, 6109)$ | $3.0320203144659480177269423190$ | $\infty$ |
Integral points
\( \left(118, 6109\right) \), \( \left(118, -6228\right) \)
Invariants
| Conductor: | $N$ | = | \( 462722 \) | = | $2 \cdot 13^{2} \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $-321991054384970906$ | = | $-1 \cdot 2 \cdot 13^{7} \cdot 37^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{12167}{26} \) | = | $2^{-1} \cdot 13^{-1} \cdot 23^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0437077725181475545887568301$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0442258625347330356220347262$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8441470072585473$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.637610420614831$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.0320203144659480177269423190$ |
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| Real period: | $\Omega$ | ≈ | $0.21157444613966908368741407633$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.5659920748694329470338401547 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.565992075 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.211574 \cdot 3.032020 \cdot 4}{1^2} \\ & \approx 2.565992075\end{aligned}$$
Modular invariants
Modular form 462722.2.a.g
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5624640 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $37$ | $1$ | $I_0^{*}$ | 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 | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 34632 = 2^{3} \cdot 3^{2} \cdot 13 \cdot 37 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 34615 & 18 \\ 34614 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 15689 & 12654 \\ 24642 & 33301 \end{array}\right),\left(\begin{array}{rr} 26195 & 4662 \\ 22311 & 5105 \end{array}\right),\left(\begin{array}{rr} 9359 & 0 \\ 0 & 34631 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 25975 & 29970 \\ 6327 & 27307 \end{array}\right),\left(\begin{array}{rr} 17317 & 29970 \\ 32301 & 27307 \end{array}\right)$.
The torsion field $K:=\Q(E[34632])$ is a degree-$1980519770750976$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/34632\Z)$.
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 | $4$ | \( 231361 = 13^{2} \cdot 37^{2} \) |
| $13$ | additive | $98$ | \( 2738 = 2 \cdot 37^{2} \) |
| $37$ | additive | $686$ | \( 338 = 2 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 462722g
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 26a3, its twist by $481$.
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.