Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-44616x+2091544\)
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(homogenize, simplify) |
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\(y^2z=x^3-44616xz^2+2091544z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-44616x+2091544\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(209, 1377\right) \) | $3.3748604734268136717152548449$ | $\infty$ |
| \( \left(182, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([209:1377:1]\) | $3.3748604734268136717152548449$ | $\infty$ |
| \([182:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(209, 1377\right) \) | $3.3748604734268136717152548449$ | $\infty$ |
| \( \left(182, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(182, 0\right) \), \((209,\pm 1377)\), \((2210,\pm 103428)\)
\([182:0:1]\), \([209:\pm 1377:1]\), \([2210:\pm 103428:1]\)
\( \left(182, 0\right) \), \((209,\pm 1377)\), \((2210,\pm 103428)\)
Invariants
| Conductor: | $N$ | = | \( 97344 \) | = | $2^{6} \cdot 3^{2} \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $3794162872660992$ | = | $2^{10} \cdot 3^{10} \cdot 13^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{2725888}{1053} \) | = | $2^{11} \cdot 3^{-4} \cdot 11^{3} \cdot 13^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6883325572875695373257476191$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.72107091624387476757964548803$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9241957238555172$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.807341303567336$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.3748604734268136717152548449$ |
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| Real period: | $\Omega$ | ≈ | $0.40256678484356577594018361344$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.4344269203322665325831496088 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.434426920 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.402567 \cdot 3.374860 \cdot 16}{2^2} \\ & \approx 5.434426920\end{aligned}$$
Modular invariants
Modular form 97344.2.a.br
For more coefficients, see the Downloads section to the right.
| Modular degree: | 344064 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | -1 | 6 | 10 | 0 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $13$ | $2$ | $I_{1}^{*}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.12.0.13 | $12$ |
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
$\left(\begin{array}{rr} 7 & 6 \\ 306 & 307 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 103 & 304 \\ 100 & 279 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 88 & 309 \\ 211 & 310 \end{array}\right),\left(\begin{array}{rr} 269 & 270 \\ 182 & 29 \end{array}\right),\left(\begin{array}{rr} 305 & 8 \\ 304 & 9 \end{array}\right),\left(\begin{array}{rr} 191 & 192 \\ 26 & 185 \end{array}\right)$.
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 | $2$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
| $3$ | additive | $8$ | \( 10816 = 2^{6} \cdot 13^{2} \) |
| $13$ | 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.br
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 312.f3, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{78}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.151613669376.23 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.25622710124544.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.102490840498176.2 | \(\Z/8\Z\) | 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 | add | add | ord | ss | ss | add | ord | ord | ss | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 3 | 1,1 | 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 | 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.