Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-12993360x+18030531050\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-12993360xz^2+18030531050z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-207893763x+1153746093438\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(8323/4, -8323/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 422370 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $39123672199296750$ | = | $2 \cdot 3^{9} \cdot 5^{3} \cdot 13^{2} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{261984288445803}{42250} \) | = | $2^{-1} \cdot 3^{3} \cdot 5^{-3} \cdot 7^{3} \cdot 11^{3} \cdot 13^{-2} \cdot 277^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5860269819940665703028943893$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.28984827590976407175194674566$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0039069659549966$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.690068911782476$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.28536210844134270151949467201$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot1\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.1414484337653708060779786880 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.141448434 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.285362 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 1.141448434\end{aligned}$$
Modular invariants
Modular form 422370.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23887872 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 4 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $4$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 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 \( 29640 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 14039 & 0 \\ 0 & 29639 \end{array}\right),\left(\begin{array}{rr} 7810 & 12483 \\ 21033 & 21832 \end{array}\right),\left(\begin{array}{rr} 23466 & 19057 \\ 8645 & 13586 \end{array}\right),\left(\begin{array}{rr} 18241 & 20292 \\ 15846 & 3193 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 7810 & 12483 \\ 12141 & 21832 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 18448 & 23389 \\ 27075 & 18752 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 29590 & 29631 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 29629 & 12 \\ 29628 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[29640])$ is a degree-$1189501363814400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/29640\Z)$.
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 | $4$ | \( 5415 = 3 \cdot 5 \cdot 19^{2} \) |
| $3$ | additive | $2$ | \( 9386 = 2 \cdot 13 \cdot 19^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 84474 = 2 \cdot 3^{2} \cdot 13 \cdot 19^{2} \) |
| $13$ | nonsplit multiplicative | $14$ | \( 32490 = 2 \cdot 3^{2} \cdot 5 \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 422370.d
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 1170.c2, its twist by $57$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.