Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+x^2-143733x+51645444\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+x^2z-143733xz^2+51645444z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-186278400x+2411805186000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-446, 5218)$ | $4.6785589503280820357331273190$ | $\infty$ |
Integral points
\( \left(-446, 5218\right) \), \( \left(-446, -5219\right) \)
Invariants
| Conductor: | $N$ | = | \( 15925 \) | = | $5^{2} \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-964348431707171875$ | = | $-1 \cdot 5^{6} \cdot 7^{15} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{178643795968}{524596891} \) | = | $-1 \cdot 2^{27} \cdot 7^{-9} \cdot 11^{3} \cdot 13^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1377778723611494936531367537$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.36010384161644265380008071536$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1502342302824948$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.068894395910515$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.6785589503280820357331273190$ |
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| Real period: | $\Omega$ | ≈ | $0.24516353978814211494151271608$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.2940241467398542979055525339 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.294024147 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.245164 \cdot 4.678559 \cdot 2}{1^2} \\ & \approx 2.294024147\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 186624 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $2$ | $I_{9}^{*}$ | additive | -1 | 2 | 15 | 9 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 \( 8190 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 5849 & 1620 \\ 225 & 6389 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 6570 \\ 0 & 911 \end{array}\right),\left(\begin{array}{rr} 4913 & 0 \\ 0 & 8189 \end{array}\right),\left(\begin{array}{rr} 1651 & 6570 \\ 5535 & 511 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 8173 & 18 \\ 8172 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[8190])$ is a degree-$4108475105280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8190\Z)$.
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 |
|---|---|---|---|
| $5$ | additive | $14$ | \( 637 = 7^{2} \cdot 13 \) |
| $7$ | additive | $32$ | \( 325 = 5^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 15925m
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 91b3, its twist by $-35$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.364.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.12057136.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.892699045875.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.24102874238625.2 | \(\Z/9\Z\) | not in database |
| $6$ | 6.2.3130218000.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.492449685436859862760816064254776000000000.1 | \(\Z/6\Z\) | not in database |
| $18$ | 18.6.9692887158453712678721142592726756008000000000.1 | \(\Z/18\Z\) | 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 | ss | ord | add | add | ss | split | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 18,3 | 5 | - | - | 1,1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0,0 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.