Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-8440x+293567\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-8440xz^2+293567z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-10938267x+13729476726\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(499, 10723\right) \) | $5.4121318688021718425153854124$ | $\infty$ |
| \( \left(-106, 53\right) \) | $0$ | $2$ |
| \( \left(58, -29\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([499:10723:1]\) | $5.4121318688021718425153854124$ | $\infty$ |
| \([-106:53:1]\) | $0$ | $2$ |
| \([58:-29:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(17967, 2370060\right) \) | $5.4121318688021718425153854124$ | $\infty$ |
| \( \left(-3813, 0\right) \) | $0$ | $2$ |
| \( \left(2091, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-106, 53\right) \), \( \left(58, -29\right) \), \( \left(499, 10723\right) \), \( \left(499, -11222\right) \)
\([-106:53:1]\), \([58:-29:1]\), \([499:10723:1]\), \([499:-11222:1]\)
\( \left(-3813, 0\right) \), \( \left(2091, 0\right) \), \((17967,\pm 2370060)\)
Invariants
| Conductor: | $N$ | = | \( 25215 \) | = | $3 \cdot 5 \cdot 41^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $1068773454225$ | = | $3^{2} \cdot 5^{2} \cdot 41^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{13997521}{225} \) | = | $3^{-2} \cdot 5^{-2} \cdot 241^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1079348709893374597811691286$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.74885116236281644215221255792$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9622954410890624$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.821912231993467$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.4121318688021718425153854124$ |
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| Real period: | $\Omega$ | ≈ | $0.87494978413483611873808000540$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.7353436103177274476276749634 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.735343610 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.874950 \cdot 5.412132 \cdot 16}{4^2} \\ & \approx 4.735343610\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 33280 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $41$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cs | 16.48.0.3 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9840 = 2^{4} \cdot 3 \cdot 5 \cdot 41 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 3281 & 7216 \\ 246 & 3937 \end{array}\right),\left(\begin{array}{rr} 8399 & 0 \\ 0 & 9839 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2461 & 3608 \\ 410 & 4757 \end{array}\right),\left(\begin{array}{rr} 6725 & 7216 \\ 6724 & 7873 \end{array}\right),\left(\begin{array}{rr} 3855 & 3608 \\ 5248 & 5249 \end{array}\right),\left(\begin{array}{rr} 9825 & 16 \\ 9824 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9840])$ is a degree-$2031353856000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9840\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 1681 = 41^{2} \) |
| $3$ | split multiplicative | $4$ | \( 8405 = 5 \cdot 41^{2} \) |
| $5$ | split multiplicative | $6$ | \( 5043 = 3 \cdot 41^{2} \) |
| $41$ | additive | $842$ | \( 15 = 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 25215h
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 15a3, its twist by $41$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{41}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.2.41.1-225.1-b5 |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{-41})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{-41})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{41})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | \(\Q(i, \sqrt{15}, \sqrt{41})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | \(\Q(i, \sqrt{3}, \sqrt{41})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.8.3576353765625.1 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.127758803665936000000000000.1 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | ord | split | split | ss | ord | ord | ord | ord | ss | ord | ss | ord | add | ord | ord |
| $\lambda$-invariant(s) | 4 | 4 | 2 | 1,1 | 1 | 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 | 0,0 | 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.