Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-10326x-370059\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-10326xz^2-370059z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-13383171x-17064728514\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-44, 61)$ | $2.3063106111971791324162210666$ | $\infty$ |
| $(-293/4, 293/8)$ | $0$ | $2$ |
Integral points
\( \left(-44, 61\right) \), \( \left(-44, -17\right) \)
Invariants
| Conductor: | $N$ | = | \( 20631 \) | = | $3 \cdot 13 \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $12684159077187$ | = | $3 \cdot 13^{4} \cdot 23^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{822656953}{85683} \) | = | $3^{-1} \cdot 13^{-4} \cdot 937^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2493806458361873924545993987$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.31836646212838745294877701721$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9608618677341092$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9600198130218636$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.3063106111971791324162210666$ |
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| Real period: | $\Omega$ | ≈ | $0.47678668913523622060328116987$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.3984728017206643811206902256 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.398472802 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.476787 \cdot 2.306311 \cdot 16}{2^2} \\ & \approx 4.398472802\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 50688 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $23$ | $4$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7176 = 2^{3} \cdot 3 \cdot 13 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7169 & 8 \\ 7168 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6280 & 5819 \\ 667 & 4440 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 7170 & 7171 \end{array}\right),\left(\begin{array}{rr} 5728 & 6555 \\ 1357 & 2186 \end{array}\right),\left(\begin{array}{rr} 5659 & 5658 \\ 4186 & 6211 \end{array}\right),\left(\begin{array}{rr} 6073 & 3128 \\ 4324 & 5337 \end{array}\right),\left(\begin{array}{rr} 1559 & 0 \\ 0 & 7175 \end{array}\right)$.
The torsion field $K:=\Q(E[7176])$ is a degree-$10754978217984$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7176\Z)$.
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 | $2$ | \( 1587 = 3 \cdot 23^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 6877 = 13 \cdot 23^{2} \) |
| $13$ | split multiplicative | $14$ | \( 1587 = 3 \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 39 = 3 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 20631a
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 39a3, its twist by $-23$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-69}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-23}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-23})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.13369611976704.32 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.18414809397504.37 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | nonsplit | ord | ord | ord | split | ord | ss | add | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 4 | 1 | 1 | 1 | 1 | 2 | 1 | 1,1 | - | 1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 1 | 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.