Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-4267150513x+107290031901057\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-4267150513xz^2+107290031901057z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-68274408203x+6866493767259462\)
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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 |
|---|---|---|
| $(38365, 200168)$ | $2.4170815208212516419492198032$ | $\infty$ |
| $(603445/16, -1219033/64)$ | $4.5920991103264560677362309077$ | $\infty$ |
| $(150859/4, -150863/8)$ | $0$ | $2$ |
Integral points
\( \left(38365, 200168\right) \), \( \left(38365, -238534\right) \), \( \left(62521, 9182174\right) \), \( \left(62521, -9244696\right) \), \( \left(1799971, 2412425984\right) \), \( \left(1799971, -2414225956\right) \)
Invariants
| Conductor: | $N$ | = | \( 413270 \) | = | $2 \cdot 5 \cdot 11 \cdot 13 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $66824377025120$ | = | $2^{5} \cdot 5 \cdot 11^{3} \cdot 13 \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{355995140004443961140387841}{2768480} \) | = | $2^{-5} \cdot 3^{3} \cdot 5^{-1} \cdot 11^{-3} \cdot 13^{-1} \cdot 236243627^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.6748097479472025147225331674$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.2582030759190944745977658585$ |
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| $abc$ quality: | $Q$ | ≈ | $1.139189521584851$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.042151790875353$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $10.767914537053080279279042948$ |
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| Real period: | $\Omega$ | ≈ | $0.13765618708300557074697367638$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 60 $ = $ 5\cdot1\cdot3\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $22.234050870095912096297608782 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 22.234050870 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.137656 \cdot 10.767915 \cdot 60}{2^2} \\ & \approx 22.234050870\end{aligned}$$
Modular invariants
Modular form 413270.2.a.ci
For more coefficients, see the Downloads section to the right.
| Modular degree: | 137625600 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $4$ | $I_0^{*}$ | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 97240 = 2^{3} \cdot 5 \cdot 11 \cdot 13 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 2296 & 34323 \\ 57205 & 11442 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 60776 & 6443 \\ 9299 & 37928 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 79204 & 11441 \\ 83623 & 57206 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 22879 & 0 \\ 0 & 97239 \end{array}\right),\left(\begin{array}{rr} 62408 & 34323 \\ 21845 & 11442 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 97234 & 97235 \end{array}\right),\left(\begin{array}{rr} 10728 & 30753 \\ 17867 & 52174 \end{array}\right),\left(\begin{array}{rr} 97233 & 8 \\ 97232 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[97240])$ is a degree-$416255915851776000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/97240\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 206635 = 5 \cdot 11 \cdot 13 \cdot 17^{2} \) |
| $3$ | good | $2$ | \( 37570 = 2 \cdot 5 \cdot 13 \cdot 17^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 41327 = 11 \cdot 13 \cdot 17^{2} \) |
| $11$ | split multiplicative | $12$ | \( 37570 = 2 \cdot 5 \cdot 13 \cdot 17^{2} \) |
| $13$ | split multiplicative | $14$ | \( 31790 = 2 \cdot 5 \cdot 11 \cdot 17^{2} \) |
| $17$ | additive | $146$ | \( 1430 = 2 \cdot 5 \cdot 11 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 413270ci
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1430h4, its twist by $17$.
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.