Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1299x+16850\)
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(homogenize, simplify) |
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\(y^2z=x^3-1299xz^2+16850z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1299x+16850\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(7, 90\right) \) | $1.6234811978893632661108716440$ | $\infty$ |
| \( \left(25, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([7:90:1]\) | $1.6234811978893632661108716440$ | $\infty$ |
| \([25:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(7, 90\right) \) | $1.6234811978893632661108716440$ | $\infty$ |
| \( \left(25, 0\right) \) | $0$ | $2$ |
Integral points
\((-7,\pm 160)\), \((7,\pm 90)\), \( \left(25, 0\right) \)
\([-7:\pm 160:1]\), \([7:\pm 90:1]\), \([25:0:1]\)
\((-7,\pm 160)\), \((7,\pm 90)\), \( \left(25, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 5904 \) | = | $2^{4} \cdot 3^{2} \cdot 41$ |
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| Minimal Discriminant: | $\Delta$ | = | $17629249536$ | = | $2^{16} \cdot 3^{8} \cdot 41 $ |
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| j-invariant: | $j$ | = | \( \frac{81182737}{5904} \) | = | $2^{-4} \cdot 3^{-2} \cdot 41^{-1} \cdot 433^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.71244829550689630163306469698$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.53000502938710385348179004294$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9582556017330255$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.814370972334694$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.6234811978893632661108716440$ |
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| Real period: | $\Omega$ | ≈ | $1.2043400770445954602547840063$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.9104469418930557692476057480 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.910446942 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.204340 \cdot 1.623481 \cdot 8}{2^2} \\ & \approx 3.910446942\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4608 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{8}^{*}$ | additive | -1 | 4 | 16 | 4 |
| $3$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $41$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.12.0.11 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 984 = 2^{3} \cdot 3 \cdot 41 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 978 & 979 \end{array}\right),\left(\begin{array}{rr} 844 & 657 \\ 15 & 334 \end{array}\right),\left(\begin{array}{rr} 787 & 126 \\ 210 & 43 \end{array}\right),\left(\begin{array}{rr} 655 & 0 \\ 0 & 983 \end{array}\right),\left(\begin{array}{rr} 119 & 120 \\ 930 & 113 \end{array}\right),\left(\begin{array}{rr} 977 & 8 \\ 976 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[984])$ is a degree-$4231987200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/984\Z)$.
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 | $2$ | \( 369 = 3^{2} \cdot 41 \) |
| $3$ | additive | $8$ | \( 656 = 2^{4} \cdot 41 \) |
| $41$ | nonsplit multiplicative | $42$ | \( 144 = 2^{4} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 5904m
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 246e1, its twist by $12$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{41}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{123}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{41})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.6156135096336.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.25215529354592256.1 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.8923447296.13 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.512588885879808.3 | \(\Z/6\Z\) | 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 | add | add | ord | ord | ord | ord | ord | ord | ss | ord | ord | ord | nonsplit | ord | ord |
| $\lambda$-invariant(s) | - | - | 3 | 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 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.