Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-236673x-44257219\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-236673xz^2-44257219z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3786771x-2836248786\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{4475}{16}, \frac{9151}{64}\right) \) | $5.9086026158768439942103117512$ | $\infty$ |
| \( \left(562, -281\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-17900:9151:64]\) | $5.9086026158768439942103117512$ | $\infty$ |
| \([562:-281:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{4479}{4}, \frac{201}{8}\right) \) | $5.9086026158768439942103117512$ | $\infty$ |
| \( \left(2247, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(562, -281\right) \)
\([562:-281:1]\)
\( \left(2247, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 14706 \) | = | $2 \cdot 3^{2} \cdot 19 \cdot 43$ |
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| Minimal Discriminant: | $\Delta$ | = | $28734995054592$ | = | $2^{14} \cdot 3^{3} \cdot 19 \cdot 43^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{54300912478267192011}{1064259076096} \) | = | $2^{-14} \cdot 3^{3} \cdot 19^{-1} \cdot 43^{-4} \cdot 199^{3} \cdot 6343^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7043398183875889497602891008$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4296867462205615269114777916$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0174801586907658$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.0788718686250345$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.9086026158768439942103117512$ |
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| Real period: | $\Omega$ | ≈ | $0.21645259679035985323661278166$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.5578647592177119903034394691 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.557864759 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.216453 \cdot 5.908603 \cdot 8}{2^2} \\ & \approx 2.557864759\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 111104 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{14}$ | nonsplit multiplicative | 1 | 1 | 14 | 14 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $19$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $43$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 456 = 2^{3} \cdot 3 \cdot 19 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 308 & 1 \\ 151 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 194 & 1 \\ 359 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 453 & 4 \\ 452 & 5 \end{array}\right),\left(\begin{array}{rr} 229 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 289 & 172 \\ 56 & 399 \end{array}\right)$.
The torsion field $K:=\Q(E[456])$ is a degree-$756449280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/456\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 57 = 3 \cdot 19 \) |
| $3$ | additive | $6$ | \( 1634 = 2 \cdot 19 \cdot 43 \) |
| $7$ | good | $2$ | \( 7353 = 3^{2} \cdot 19 \cdot 43 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 774 = 2 \cdot 3^{2} \cdot 43 \) |
| $43$ | nonsplit multiplicative | $44$ | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 14706.b
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{57}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.4.32832.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.35119561982976.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.389136420864.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | add | ord | ss | ord | ss | ss | nonsplit | ord | ord | ord | ord | ord | nonsplit | ord |
| $\lambda$-invariant(s) | 2 | - | 1 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.