Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+13689x-1542294\)
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(homogenize, simplify) |
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\(y^2z=x^3+13689xz^2-1542294z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+13689x-1542294\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(429, 9126\right) \) | $1.1510033699329495496659500860$ | $\infty$ |
| \( \left(91, 676\right) \) | $1.9865797511014959296553800846$ | $\infty$ |
| \( \left(78, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([429:9126:1]\) | $1.1510033699329495496659500860$ | $\infty$ |
| \([91:676:1]\) | $1.9865797511014959296553800846$ | $\infty$ |
| \([78:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(429, 9126\right) \) | $1.1510033699329495496659500860$ | $\infty$ |
| \( \left(91, 676\right) \) | $1.9865797511014959296553800846$ | $\infty$ |
| \( \left(78, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(78, 0\right) \), \((91,\pm 676)\), \((103,\pm 980)\), \((105,\pm 1026)\), \((169,\pm 2366)\), \((321,\pm 5994)\), \((429,\pm 9126)\), \((1261,\pm 44954)\), \((2535,\pm 127764)\), \((18850,\pm 2588066)\), \((1005801,\pm 1008714114)\)
\([78:0:1]\), \([91:\pm 676:1]\), \([103:\pm 980:1]\), \([105:\pm 1026:1]\), \([169:\pm 2366:1]\), \([321:\pm 5994:1]\), \([429:\pm 9126:1]\), \([1261:\pm 44954:1]\), \([2535:\pm 127764:1]\), \([18850:\pm 2588066:1]\), \([1005801:\pm 1008714114:1]\)
\( \left(78, 0\right) \), \((91,\pm 676)\), \((103,\pm 980)\), \((105,\pm 1026)\), \((169,\pm 2366)\), \((321,\pm 5994)\), \((429,\pm 9126)\), \((1261,\pm 44954)\), \((2535,\pm 127764)\), \((18850,\pm 2588066)\), \((1005801,\pm 1008714114)\)
Invariants
| Conductor: | $N$ | = | \( 85176 \) | = | $2^{3} \cdot 3^{2} \cdot 7 \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-1191756286925568$ | = | $-1 \cdot 2^{8} \cdot 3^{9} \cdot 7^{2} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{11664}{49} \) | = | $2^{4} \cdot 3^{6} \cdot 7^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5754329449315418594888698794$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.99309907067360565002912918338$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8374478418484275$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.701457924670551$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.2565964382040631295444243413$ |
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| Real period: | $\Omega$ | ≈ | $0.24626330314192750139677537342$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $8.8914702836870570949966772155 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.891470284 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.246263 \cdot 2.256596 \cdot 64}{2^2} \\ & \approx 8.891470284\end{aligned}$$
Modular invariants
Modular form 85176.2.a.l
For more coefficients, see the Downloads section to the right.
| Modular degree: | 368640 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{1}^{*}$ | additive | -1 | 3 | 8 | 0 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_0^{*}$ | 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 |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 84 = 2^{2} \cdot 3 \cdot 7 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 81 & 4 \\ 80 & 5 \end{array}\right),\left(\begin{array}{rr} 32 & 1 \\ 55 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 65 & 22 \\ 20 & 63 \end{array}\right),\left(\begin{array}{rr} 73 & 4 \\ 62 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[84])$ is a degree-$774144$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/84\Z)$.
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 | $2$ | \( 507 = 3 \cdot 13^{2} \) |
| $3$ | additive | $2$ | \( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 12168 = 2^{3} \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 85176.l
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 504.d2, its twist by $-39$.
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 |
| $4$ | 4.2.3577392.3 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.261178235136.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.12797733521664.82 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | add | ord | nonsplit | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 4 | 2 | 2 | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | - | 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.