Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-692x+6941\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-692xz^2+6941z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-896211x+326539566\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(51, 298)$ | $2.7103794096947463444162522068$ | $\infty$ |
| $(15, -8)$ | $0$ | $2$ |
Integral points
\( \left(15, -8\right) \), \( \left(51, 298\right) \), \( \left(51, -350\right) \)
Invariants
| Conductor: | $N$ | = | \( 100011 \) | = | $3 \cdot 17 \cdot 37 \cdot 53$ |
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| Discriminant: | $\Delta$ | = | $2700297$ | = | $3^{4} \cdot 17 \cdot 37 \cdot 53 $ |
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| j-invariant: | $j$ | = | \( \frac{36571225840057}{2700297} \) | = | $3^{-4} \cdot 17^{-1} \cdot 19^{3} \cdot 37^{-1} \cdot 53^{-1} \cdot 1747^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.28428927929730370921236036577$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.28428927929730370921236036577$ |
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| $abc$ quality: | $Q$ | ≈ | $0.816876179885067$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.712601987489169$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.7103794096947463444162522068$ |
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| Real period: | $\Omega$ | ≈ | $2.4332083556482169414986080414$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2^{2}\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.5949178266461386560924636555 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.594917827 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.433208 \cdot 2.710379 \cdot 4}{2^2} \\ & \approx 6.594917827\end{aligned}$$
Modular invariants
Modular form 100011.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 45312 |
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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 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 |
| $17$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $37$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $53$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 800088 = 2^{3} \cdot 3 \cdot 17 \cdot 37 \cdot 53 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 470646 & 1 \\ 470663 & 4 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 256638 & 1 \\ 317039 & 4 \end{array}\right),\left(\begin{array}{rr} 108126 & 1 \\ 281135 & 4 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 800081 & 8 \\ 800080 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 28 & 75 \end{array}\right),\left(\begin{array}{rr} 400045 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 200023 & 8 \\ 400044 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 533393 & 8 \\ 533396 & 33 \end{array}\right)$.
The torsion field $K:=\Q(E[800088])$ is a degree-$1696755424900109303808$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/800088\Z)$.
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 | $2$ | \( 33337 = 17 \cdot 37 \cdot 53 \) |
| $3$ | split multiplicative | $4$ | \( 33337 = 17 \cdot 37 \cdot 53 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 5883 = 3 \cdot 37 \cdot 53 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 2703 = 3 \cdot 17 \cdot 53 \) |
| $53$ | split multiplicative | $54$ | \( 1887 = 3 \cdot 17 \cdot 37 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 100011d
consists of 2 curves linked by isogenies of
degree 2.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33337}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.533392.1 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 53 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ord | split | ord | ord | ord | ord | nonsplit | ord | ord | ord | ord | nonsplit | ord | ord | ord | split |
| $\lambda$-invariant(s) | 3 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.