Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-313793x-376531905\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-313793xz^2-376531905z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-25417260x-274415506992\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(865, 0)$ | $0$ | $2$ |
Integral points
\( \left(865, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 23232 \) | = | $2^{6} \cdot 3 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $-59235852123874787328$ | = | $-1 \cdot 2^{21} \cdot 3^{2} \cdot 11^{12} $ |
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| j-invariant: | $j$ | = | \( -\frac{7357983625}{127552392} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-2} \cdot 5^{3} \cdot 11^{-6} \cdot 389^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4751650233637961293025566793$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.23649661612469289314573670813$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0528666156799333$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.273461532390553$ | |||
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: | $\Omega$ | ≈ | $0.084970234853693423416859079545$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.0589284547329632430069268636 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.058928455 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.084970 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 3.058928455\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 552960 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{11}^{*}$ | additive | -1 | 6 | 21 | 3 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
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 | 8.6.0.5 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 254 & 261 \\ 159 & 8 \end{array}\right),\left(\begin{array}{rr} 253 & 12 \\ 252 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 119 & 252 \\ 186 & 191 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 214 & 255 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 54 & 263 \\ 187 & 142 \end{array}\right),\left(\begin{array}{rr} 89 & 12 \\ 226 & 73 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$10137600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 | $4$ | \( 121 = 11^{2} \) |
| $3$ | split multiplicative | $4$ | \( 7744 = 2^{6} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 192 = 2^{6} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 23232dm
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 66a4, its twist by $88$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-66}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.34848.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{33})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.40240240128.4 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.77720518656.13 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.552679243776.17 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.22163535035015424154317684538909051060224.2 | \(\Z/18\Z\) | 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 | 11 |
|---|---|---|---|
| Reduction type | add | split | add |
| $\lambda$-invariant(s) | - | 3 | - |
| $\mu$-invariant(s) | - | 1 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
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 $0$.