Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-1891302x+1000511460\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-1891302xz^2+1000511460z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2451127419x+46687216059990\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1336, 34478)$ | $5.6826595858291772067561767741$ | $\infty$ |
| $(-1588, 794)$ | $0$ | $2$ |
| $(780, -390)$ | $0$ | $2$ |
Integral points
\( \left(-1588, 794\right) \), \( \left(-1336, 34478\right) \), \( \left(-1336, -33142\right) \), \( \left(780, -390\right) \)
Invariants
| Conductor: | $N$ | = | \( 57498 \) | = | $2 \cdot 3 \cdot 7 \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $396544880429865216$ | = | $2^{8} \cdot 3^{2} \cdot 7^{2} \cdot 37^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{291605712526297}{154554624} \) | = | $2^{-8} \cdot 3^{-2} \cdot 7^{-2} \cdot 13^{3} \cdot 37^{-2} \cdot 5101^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3266777133179373057914021892$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.52121875699582508360735435369$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9488068235443301$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.015913488528987$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.6826595858291772067561767741$ |
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| Real period: | $\Omega$ | ≈ | $0.29608080894281286830434644455$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{3}\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $13.460211576951461768514782714 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 13.460211577 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.296081 \cdot 5.682660 \cdot 128}{4^2} \\ & \approx 13.460211577\end{aligned}$$
Modular invariants
Modular form 57498.2.a.t
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1050624 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $37$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3108 = 2^{2} \cdot 3 \cdot 7 \cdot 37 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 923 & 3104 \\ 1846 & 3099 \end{array}\right),\left(\begin{array}{rr} 3105 & 4 \\ 3104 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 2 \\ 1034 & 3107 \end{array}\right),\left(\begin{array}{rr} 1555 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1333 & 4 \\ 2666 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[3108])$ is a degree-$352656654336$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3108\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 1369 = 37^{2} \) |
| $3$ | split multiplicative | $4$ | \( 19166 = 2 \cdot 7 \cdot 37^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 8214 = 2 \cdot 3 \cdot 37^{2} \) |
| $37$ | additive | $722$ | \( 42 = 2 \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 57498.t
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1554.c2, its twist by $37$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-3}, \sqrt{37})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{111})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-37})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | split | split | ord | nonsplit | ss | ord | ord | ord | ss | ord | ss | add | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 2 | 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 | 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.