Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-2369089x+1441050911\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-2369089xz^2+1441050911z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-191896236x+1051101802800\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-325, 46656)$ | $0.45782139261872102158333417971$ | $\infty$ |
| $(-1783, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1783, 0\right) \), \((-325,\pm 46656)\), \((674,\pm 12285)\), \((955,\pm 7104)\), \((1133,\pm 14580)\), \((3131,\pm 157248)\)
Invariants
| Conductor: | $N$ | = | \( 122304 \) | = | $2^{6} \cdot 3 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-47097008065584562176$ | = | $-1 \cdot 2^{21} \cdot 3^{18} \cdot 7^{3} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{16354376146655191}{523792501128} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-18} \cdot 13^{-2} \cdot 41^{6} \cdot 151^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5506817853152578227549635814$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0244834772115115323527772134$ |
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| $abc$ quality: | $Q$ | ≈ | $1.2004354498689798$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.755016048741616$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.45782139261872102158333417971$ |
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| Real period: | $\Omega$ | ≈ | $0.20050716270787115532746383070$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 288 $ = $ 2^{2}\cdot( 2 \cdot 3^{2} )\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.6093457291881161928521438913 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.609345729 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.200507 \cdot 0.457821 \cdot 288}{2^2} \\ & \approx 6.609345729\end{aligned}$$
Modular invariants
Modular form 122304.2.a.fd
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3317760 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{11}^{*}$ | additive | 1 | 6 | 21 | 3 |
| $3$ | $18$ | $I_{18}$ | split multiplicative | -1 | 1 | 18 | 18 |
| $7$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 2017 & 4 \\ 1850 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2181 & 4 \\ 2180 & 5 \end{array}\right),\left(\begin{array}{rr} 1457 & 4 \\ 730 & 9 \end{array}\right),\left(\begin{array}{rr} 628 & 1 \\ 935 & 0 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 1091 & 0 \end{array}\right),\left(\begin{array}{rr} 1913 & 274 \\ 272 & 1911 \end{array}\right)$.
The torsion field $K:=\Q(E[2184])$ is a degree-$324620255232$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2184\Z)$.
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 | $4$ | \( 7 \) |
| $3$ | split multiplicative | $4$ | \( 40768 = 2^{6} \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $20$ | \( 2496 = 2^{6} \cdot 3 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 122304dm
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 3822w2, its twist by $8$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-14}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.16694496.2 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.17837196588417024.103 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | split | ord | add | ord | nonsplit | ord | ord | ord | ss | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | - | 6 | 3 | - | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1,3 | 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.