Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-91517x-10205499\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-91517xz^2-10205499z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1464267x-654616186\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-201, 296)$ | $1.6722520697130763885537848051$ | $\infty$ |
| $(-145, 72)$ | $0$ | $2$ |
Integral points
\( \left(-201, 296\right) \), \( \left(-201, -96\right) \), \( \left(-145, 72\right) \), \( \left(359, 1584\right) \), \( \left(359, -1944\right) \), \( \left(6255, 490952\right) \), \( \left(6255, -497208\right) \)
Invariants
| Conductor: | $N$ | = | \( 57330 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $3859008877301760$ | = | $2^{12} \cdot 3^{6} \cdot 5 \cdot 7^{6} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{988345570681}{44994560} \) | = | $2^{-12} \cdot 5^{-1} \cdot 7^{3} \cdot 13^{-3} \cdot 1423^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7527106213393750838401354349$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.23044940247766358558983644472$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9543225417743602$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.188025432289438$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.6722520697130763885537848051$ |
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| Real period: | $\Omega$ | ≈ | $0.27526215613721560362991168949$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ ( 2^{2} \cdot 3 )\cdot2\cdot1\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.047385047539426846553379221 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.047385048 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.275262 \cdot 1.672252 \cdot 96}{2^2} \\ & \approx 11.047385048\end{aligned}$$
Modular invariants
Modular form 57330.2.a.fg
For more coefficients, see the Downloads section to the right.
| Modular degree: | 497664 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 | 4.6.0.3 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 3858 & 1561 \\ 4823 & 8 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 24 \\ 492 & 1687 \end{array}\right),\left(\begin{array}{rr} 3119 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 21 & 4 \\ 10820 & 10901 \end{array}\right),\left(\begin{array}{rr} 10312 & 4683 \\ 2037 & 6154 \end{array}\right),\left(\begin{array}{rr} 5461 & 4704 \\ 5460 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 7279 & 6216 \\ 7280 & 10919 \end{array}\right),\left(\begin{array}{rr} 10897 & 24 \\ 10896 & 25 \end{array}\right),\left(\begin{array}{rr} 8191 & 4704 \\ 7812 & 1849 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$4869303828480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 28665 = 3^{2} \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $3$ | additive | $6$ | \( 245 = 5 \cdot 7^{2} \) |
| $5$ | split multiplicative | $6$ | \( 11466 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 57330ez
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 130a3, its twist by $21$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.458640.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{65})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.468838125.1 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.888731494560000.54 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.210350649600.9 | \(\Z/12\Z\) | 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 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.3823624125622766342690786032633828800000000.2 | \(\Z/18\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | add | split | add | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 8 | - | 2 | - | 1 | 1 | 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.