Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-1557845383x-23653796228447\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-1557845383xz^2-23653796228447z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2018967617043x-1103561232320170962\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-92917/4, 92917/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 35490 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $273635311017579486321904740$ | = | $2^{2} \cdot 3^{8} \cdot 5 \cdot 7^{16} \cdot 13^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{86623684689189325642735681}{56690726941459561860} \) | = | $2^{-2} \cdot 3^{-8} \cdot 5^{-1} \cdot 7^{-16} \cdot 13^{-1} \cdot 23^{3} \cdot 4007^{3} \cdot 4801^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.0124729709612045816723914786$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.7299982922304362136456477578$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0354092118359115$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.169348751826162$ | |||
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: | $\Omega$ | ≈ | $0.024031757182921275669038658070$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2\cdot2\cdot1\cdot2^{4}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.76901622985348082140923705825 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.769016230 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.024032 \cdot 1.000000 \cdot 128}{2^2} \\ & \approx 0.769016230\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 22020096 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
| $13$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
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 | 16.48.0.121 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 14560 = 2^{5} \cdot 5 \cdot 7 \cdot 13 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 4161 & 32 \\ 8336 & 513 \end{array}\right),\left(\begin{array}{rr} 12741 & 32 \\ 12756 & 513 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 11998 & 12555 \end{array}\right),\left(\begin{array}{rr} 2934 & 3 \\ 13005 & 14348 \end{array}\right),\left(\begin{array}{rr} 8191 & 32 \\ 4550 & 1 \end{array}\right),\left(\begin{array}{rr} 14529 & 32 \\ 14528 & 33 \end{array}\right),\left(\begin{array}{rr} 5584 & 14535 \\ 13641 & 11374 \end{array}\right)$.
The torsion field $K:=\Q(E[14560])$ is a degree-$12984810209280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/14560\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 845 = 5 \cdot 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 11830 = 2 \cdot 5 \cdot 7 \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \) |
| $7$ | split multiplicative | $8$ | \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 35490k
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 2730v7, its twist by $13$.
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 |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-13}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{-13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-13})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-10}, \sqrt{-13})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.4.308915776000000.14 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.19307236000000.34 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.1169858560000.4 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.1265319018496000000.6 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.1265319018496000000.9 | \(\Z/16\Z\) | 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 | \(\Z/16\Z\) | not in database |
| $16$ | 16.0.1601032218567680790102016000000000000.32 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | 13 |
|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | nonsplit | split | add |
| $\lambda$-invariant(s) | 4 | 0 | 0 | 5 | - |
| $\mu$-invariant(s) | 3 | 0 | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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 $0$.