Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-22780524x-41849826928\)
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(homogenize, simplify) |
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\(y^2z=x^3-22780524xz^2-41849826928z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-22780524x-41849826928\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2756, 0)$ | $0$ | $2$ |
Integral points
\( \left(-2756, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 97344 \) | = | $2^{6} \cdot 3^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $40471070641717248$ | = | $2^{15} \cdot 3^{9} \cdot 13^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{11339065490696}{351} \) | = | $2^{3} \cdot 3^{-3} \cdot 13^{-1} \cdot 47^{3} \cdot 239^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6910707556941772388285653488$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.0071440430705776116673411422648$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0110791802338388$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.435993203026314$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.069104865828460620634511553544$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.9755503396491646856848318551 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 4.975550340 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.069105 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 4.975550340\end{aligned}$$
Modular invariants
Modular form 97344.2.a.fd
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4128768 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{5}^{*}$ | additive | 1 | 6 | 15 | 0 |
| $3$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $13$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 | 8.12.0.14 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 312 = 2^{3} \cdot 3 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 44 & 311 \\ 97 & 306 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 306 & 307 \end{array}\right),\left(\begin{array}{rr} 192 & 31 \\ 125 & 138 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 200 & 309 \\ 203 & 310 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 192 & 31 \\ 47 & 60 \end{array}\right),\left(\begin{array}{rr} 305 & 8 \\ 304 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[312])$ is a degree-$40255488$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/312\Z)$.
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 | $4$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
| $3$ | additive | $6$ | \( 10816 = 2^{6} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 576 = 2^{6} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 97344.fd
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1248.b1, its twist by $312$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{78}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-26}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-26})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.59034724126949376.9 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1639853447970816.67 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.87329473560576.39 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 13 |
|---|---|---|---|
| Reduction type | add | add | add |
| $\lambda$-invariant(s) | - | - | - |
| $\mu$-invariant(s) | - | - | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.