Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+5037837x+49175168434\)
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(homogenize, simplify) |
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\(y^2z=x^3+5037837xz^2+49175168434z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+5037837x+49175168434\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(10185, 1075648)$ | $1.5172830539516035834634405844$ | $\infty$ |
Integral points
\((1239,\pm 239414)\), \((10185,\pm 1075648)\)
Invariants
| Conductor: | $N$ | = | \( 91728 \) | = | $2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-1052844181741137031593984$ | = | $-1 \cdot 2^{19} \cdot 3^{13} \cdot 7^{13} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{40251338884511}{2997011332224} \) | = | $2^{-7} \cdot 3^{-7} \cdot 7^{-7} \cdot 13^{-1} \cdot 43^{3} \cdot 797^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2890009989207165802398486036$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0735925994990597725723174920$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0387829692161805$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.492495441268786$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.5172830539516035834634405844$ |
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| Real period: | $\Omega$ | ≈ | $0.066790011064183827250530126118$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.4857185254992787348575437033 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.485718525 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.066790 \cdot 1.517283 \cdot 64}{1^2} \\ & \approx 6.485718525\end{aligned}$$
Modular invariants
Modular form 91728.2.a.cr
For more coefficients, see the Downloads section to the right.
| Modular degree: | 10838016 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{11}^{*}$ | additive | -1 | 4 | 19 | 7 |
| $3$ | $4$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $7$ | $4$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $13$ | $1$ | $I_{1}$ | split 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 |
|---|---|---|
| $7$ | 7B.6.1 | 7.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 2171 & 14 \\ 2170 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 545 & 2170 \\ 1631 & 2085 \end{array}\right),\left(\begin{array}{rr} 727 & 2170 \\ 721 & 2085 \end{array}\right),\left(\begin{array}{rr} 1093 & 14 \\ 1099 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1637 & 1078 \\ 539 & 1383 \end{array}\right),\left(\begin{array}{rr} 2017 & 14 \\ 1015 & 99 \end{array}\right)$.
The torsion field $K:=\Q(E[2184])$ is a degree-$40577531904$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2184\Z)$.
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 | $4$ | \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \) |
| $3$ | additive | $8$ | \( 10192 = 2^{4} \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 91728.cr
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 546.f2, its twist by $-84$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-21}) \) | \(\Z/7\Z\) | not in database |
| $3$ | 3.1.2184.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.10417365504.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.1602671616.1 | \(\Z/14\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $16$ | deg 16 | \(\Z/21\Z\) | 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 | add | ord | split | ord | ord | ord | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | - | - | 1 | - | 1 | 2 | 1 | 1 | 3 | 1 | 1 | 1 | 1,1 | 1 | 1 |
| $\mu$-invariant(s) | - | - | 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.