Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-90033792x+328834457524\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-90033792xz^2+328834457524z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7292737179x+239742197746506\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(8143, 368382)$ | $6.6308363567177593482899526278$ | $\infty$ |
Integral points
\((8143,\pm 368382)\)
Invariants
| Conductor: | $N$ | = | \( 132496 \) | = | $2^{4} \cdot 7^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-13020803427034841890816$ | = | $-1 \cdot 2^{14} \cdot 7^{8} \cdot 13^{10} $ |
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| j-invariant: | $j$ | = | \( -\frac{1214950633}{196} \) | = | $-1 \cdot 2^{-2} \cdot 7^{-2} \cdot 13^{2} \cdot 193^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2528802483408479205580792451$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.55067980463136798812306878272$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9497746365910316$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.6434824441855955$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.6308363567177593482899526278$ |
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| Real period: | $\Omega$ | ≈ | $0.12199727819944165352200142680$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.2357759508218744878720871733 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.235775951 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.121997 \cdot 6.630836 \cdot 4}{1^2} \\ & \approx 3.235775951\end{aligned}$$
Modular invariants
Modular form 132496.2.a.i
For more coefficients, see the Downloads section to the right.
| Modular degree: | 21565440 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}^{*}$ | additive | -1 | 4 | 14 | 2 |
| $7$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $13$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 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 |
|---|---|---|
| $2$ | 2G | 4.2.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1092 = 2^{2} \cdot 3 \cdot 7 \cdot 13 \), index $32$, genus $0$, and generators
$\left(\begin{array}{rr} 545 & 156 \\ 819 & 1091 \end{array}\right),\left(\begin{array}{rr} 935 & 663 \\ 117 & 896 \end{array}\right),\left(\begin{array}{rr} 1081 & 12 \\ 1080 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 1054 & 1077 \end{array}\right),\left(\begin{array}{rr} 4 & 9 \\ 3 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 167 & 0 \\ 0 & 1091 \end{array}\right),\left(\begin{array}{rr} 989 & 260 \\ 0 & 781 \end{array}\right)$.
The torsion field $K:=\Q(E[1092])$ is a degree-$7608287232$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1092\Z)$.
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 | $2$ | \( 8281 = 7^{2} \cdot 13^{2} \) |
| $7$ | additive | $32$ | \( 2704 = 2^{4} \cdot 13^{2} \) |
| $13$ | additive | $50$ | \( 784 = 2^{4} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 132496.i
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 2366.a1, its twist by $364$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{91}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.676.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1827904.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.43133101869312.2 | \(\Z/3\Z\) | not in database |
| $6$ | 6.2.8150623936.2 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.8058742784152414137275306115640497688018944.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.80247603805002955388200985554312734179328.2 | \(\Z/6\Z\) | 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 | ord | ord | add | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 3 | 1 | - | 3 | - | 3 | 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 |
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.