Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-259x+1521\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-259xz^2+1521z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-21006x+1171800\)
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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 |
|---|---|---|
| $(8, 5)$ | $1.2456144929892981637071125009$ | $\infty$ |
| $(0, 39)$ | $2.5057583273478545497112642327$ | $\infty$ |
| $(9, 0)$ | $0$ | $2$ |
Integral points
\((0,\pm 39)\), \((7,\pm 10)\), \((8,\pm 5)\), \( \left(9, 0\right) \), \((41,\pm 248)\), \((209,\pm 3020)\)
Invariants
| Conductor: | $N$ | = | \( 24832 \) | = | $2^{8} \cdot 97$ |
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| Discriminant: | $\Delta$ | = | $49664$ | = | $2^{9} \cdot 97 $ |
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| j-invariant: | $j$ | = | \( \frac{3767287616}{97} \) | = | $2^{6} \cdot 97^{-1} \cdot 389^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.0080062087909230087962542675849$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.51185417662903597326666982351$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9260511127952559$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.795282374360047$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.0654751800261317266112772995$ |
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| Real period: | $\Omega$ | ≈ | $3.3087979398694133571041575335$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $5.0715189801956418452699483243 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.071518980 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 3.308798 \cdot 3.065475 \cdot 2}{2^2} \\ & \approx 5.071518980\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3456 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | -1 | 8 | 9 | 0 |
| $97$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $2$ | 2B | 8.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 776 = 2^{3} \cdot 97 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 773 & 4 \\ 772 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 1 \\ 387 & 0 \end{array}\right),\left(\begin{array}{rr} 10 & 1 \\ 383 & 0 \end{array}\right),\left(\begin{array}{rr} 681 & 98 \\ 96 & 679 \end{array}\right)$.
The torsion field $K:=\Q(E[776])$ is a degree-$11213733888$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/776\Z)$.
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 | $4$ | \( 97 \) |
| $97$ | nonsplit multiplicative | $98$ | \( 256 = 2^{8} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 24832d
consists of 2 curves linked by isogenies of
degree 2.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{194}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.4.198656.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.55899804994587590656.6 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.371318717415424.1 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | 97 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | ord | ss | ord | ord | ord | ord | ord | ord | ss | ord | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | - | 2 | 2 | 2,2 | 4 | 2 | 2 | 2 | 4 | 2 | 2,2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 0 | 0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.