Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-177712x-24856701\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-177712xz^2-24856701z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2843387x-1593672234\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(2767, 142361\right) \) | $0.99850829578801334734588555080$ | $\infty$ |
| \( \left(-\frac{1253}{4}, \frac{1249}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([2767:142361:1]\) | $0.99850829578801334734588555080$ | $\infty$ |
| \([-2506:1249:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(11067, 1149960\right) \) | $0.99850829578801334734588555080$ | $\infty$ |
| \( \left(-1254, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-203, 1781\right) \), \( \left(-203, -1579\right) \), \( \left(2767, 142361\right) \), \( \left(2767, -145129\right) \)
\([-203:1781:1]\), \([-203:-1579:1]\), \([2767:142361:1]\), \([2767:-145129:1]\)
\((-813,\pm 13440)\), \((11067,\pm 1149960)\)
Invariants
| Conductor: | $N$ | = | \( 4810 \) | = | $2 \cdot 5 \cdot 13 \cdot 37$ |
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| Minimal Discriminant: | $\Delta$ | = | $91324465801946000$ | = | $2^{4} \cdot 5^{3} \cdot 13 \cdot 37^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{620685621178022563281}{91324465801946000} \) | = | $2^{-4} \cdot 3^{3} \cdot 5^{-3} \cdot 13^{-1} \cdot 37^{-8} \cdot 59^{3} \cdot 48193^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9792352545923317075881601250$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.9792352545923317075881601250$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0170274675359632$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.646945259717884$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.99850829578801334734588555080$ |
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| Real period: | $\Omega$ | ≈ | $0.23481282392662884292674726145$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2^{2}\cdot3\cdot1\cdot2^{3} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.6271012635555762503385293874 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.627101264 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.234813 \cdot 0.998508 \cdot 96}{2^2} \\ & \approx 5.627101264\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 36864 |
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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 semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $37$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
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 | 8.24.0.57 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 38480 = 2^{4} \cdot 5 \cdot 13 \cdot 37 \), index $192$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19247 & 19250 \\ 28834 & 28823 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 8 & 65 \end{array}\right),\left(\begin{array}{rr} 7 & 19256 \\ 4666 & 23721 \end{array}\right),\left(\begin{array}{rr} 29121 & 16 \\ 2088 & 129 \end{array}\right),\left(\begin{array}{rr} 30798 & 3 \\ 23181 & 20 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 32568 & 9 \\ 11863 & 26 \end{array}\right),\left(\begin{array}{rr} 15 & 166 \\ 38194 & 35315 \end{array}\right),\left(\begin{array}{rr} 38465 & 16 \\ 38464 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[38480])$ is a degree-$2934103364075520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/38480\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 65 = 5 \cdot 13 \) |
| $3$ | good | $2$ | \( 962 = 2 \cdot 13 \cdot 37 \) |
| $5$ | split multiplicative | $6$ | \( 962 = 2 \cdot 13 \cdot 37 \) |
| $13$ | split multiplicative | $14$ | \( 370 = 2 \cdot 5 \cdot 37 \) |
| $37$ | split multiplicative | $38$ | \( 130 = 2 \cdot 5 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 4810h
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-65}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{65})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.1098500.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.16640.2 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.19307236000000.13 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.4942652416000000.45 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.1169858560000.13 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.1873048707936432.1 | \(\Z/6\Z\) | 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 |
| $16$ | deg 16 | \(\Z/16\Z\) | 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 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | ss | split | ss | ss | split | ord | ss | ord | ord | ord | split | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 1,1 | 4 | 1,1 | 1,1 | 2 | 1 | 1,1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 |
| $\mu$-invariant(s) | 2 | 0,0 | 0 | 0,0 | 0,0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.