Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-2569179x-1585192822\)
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(homogenize, simplify) |
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\(y^2z=x^3-2569179xz^2-1585192822z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2569179x-1585192822\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 101232 \) | = | $2^{4} \cdot 3^{2} \cdot 19 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $-212133255005601792$ | = | $-1 \cdot 2^{19} \cdot 3^{13} \cdot 19^{3} \cdot 37 $ |
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| j-invariant: | $j$ | = | \( -\frac{628086308429730457}{71042997888} \) | = | $-1 \cdot 2^{-7} \cdot 3^{-7} \cdot 19^{-3} \cdot 37^{-1} \cdot 856393^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3525050630054797904971004921$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1100517381114796353822457522$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9643098235711661$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.849480605644193$ | |||
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: | $\Omega$ | ≈ | $0.059623576153014081906671752562$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot2\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.71548291383616898288006103075 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.715482914 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.059624 \cdot 1.000000 \cdot 12}{1^2} \\ & \approx 0.715482914\end{aligned}$$
Modular invariants
Modular form 101232.2.a.y
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1580544 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{11}^{*}$ | additive | -1 | 4 | 19 | 7 |
| $3$ | $2$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $19$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $37$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 16872 = 2^{3} \cdot 3 \cdot 19 \cdot 37 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 8437 & 2 \\ 8437 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 15505 & 2 \\ 15505 & 3 \end{array}\right),\left(\begin{array}{rr} 16871 & 2 \\ 16870 & 3 \end{array}\right),\left(\begin{array}{rr} 13321 & 2 \\ 13321 & 3 \end{array}\right),\left(\begin{array}{rr} 12655 & 2 \\ 12655 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 16871 & 0 \end{array}\right),\left(\begin{array}{rr} 11249 & 2 \\ 11249 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[16872])$ is a degree-$8270302339399680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/16872\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 6327 = 3^{2} \cdot 19 \cdot 37 \) |
| $3$ | additive | $8$ | \( 592 = 2^{4} \cdot 37 \) |
| $19$ | split multiplicative | $20$ | \( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 101232bi consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 4218c1, its twist by $12$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.16872.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.4802857486848.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.339970046635008.4 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | ord | ord | ord | ord | ord | split | ord | ord | ord | nonsplit | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.