Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-2302105x+1343724473\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-2302105xz^2+1343724473z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2983527459x+62701759606302\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-497, 48881)$ | $5.3155092494312163761670847298$ | $\infty$ |
| $(3559/4, -3563/8)$ | $0$ | $2$ |
Integral points
\( \left(-497, 48881\right) \), \( \left(-497, -48385\right) \)
Invariants
| Conductor: | $N$ | = | \( 98553 \) | = | $3 \cdot 7 \cdot 13 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $587497162021265223$ | = | $3^{4} \cdot 7 \cdot 13^{2} \cdot 19^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{28679872714374673}{12487749183} \) | = | $3^{-4} \cdot 7^{-1} \cdot 11^{3} \cdot 13^{-2} \cdot 19^{-4} \cdot 27827^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3690638812554342855907981783$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.89684439167221405558628446236$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9282108088402934$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.832137396977355$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.3155092494312163761670847298$ |
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| Real period: | $\Omega$ | ≈ | $0.28571800243622680493683510887$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot1\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $12.149893477430195167357711129 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.149893477 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.285718 \cdot 5.315509 \cdot 32}{2^2} \\ & \approx 12.149893477\end{aligned}$$
Modular invariants
Modular form 98553.2.a.bb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1658880 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $19$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3185 & 8 \\ 3184 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 403 & 402 \\ 1210 & 2803 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 920 & 3 \\ 1373 & 2 \end{array}\right),\left(\begin{array}{rr} 1200 & 2801 \\ 1225 & 1272 \end{array}\right),\left(\begin{array}{rr} 2183 & 3184 \\ 2348 & 3159 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3186 & 3187 \end{array}\right),\left(\begin{array}{rr} 2129 & 8 \\ 2132 & 33 \end{array}\right)$.
The torsion field $K:=\Q(E[3192])$ is a degree-$381250437120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3192\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 2527 = 7 \cdot 19^{2} \) |
| $3$ | split multiplicative | $4$ | \( 32851 = 7 \cdot 13 \cdot 19^{2} \) |
| $7$ | split multiplicative | $8$ | \( 14079 = 3 \cdot 13 \cdot 19^{2} \) |
| $13$ | split multiplicative | $14$ | \( 7581 = 3 \cdot 7 \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 273 = 3 \cdot 7 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 98553bc
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 5187b3, its twist by $-19$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{19}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{133}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{19})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.19870447386384.5 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | ord | split | ord | split | ss | split | ord | add | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 4 | 2 | 3 | 2 | 1,1 | 2 | 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 | 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.