Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-82134792x+286529921991\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-82134792xz^2+286529921991z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1314156675x+18336600850750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{20931}{4}, -\frac{20931}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([41862:-20931:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(20930, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 38025 \) | = | $3^{2} \cdot 5^{2} \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $22267050362578125$ | = | $3^{10} \cdot 5^{7} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{1114544804970241}{405} \) | = | $3^{-4} \cdot 5^{-1} \cdot 103681^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9273693877589749229983816302$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.29086960847710152197363562434$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0735374703334082$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.285342819110829$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.22861855956766531771418933089$ |
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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)$ | ≈ | $0.91447423827066127085675732354 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.914474238 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.228619 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 0.914474238\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1769472 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 | 16.48.0.121 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 2341 & 2912 \\ 3796 & 2913 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1951 & 2912 \\ 390 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 479 & 0 \\ 0 & 6239 \end{array}\right),\left(\begin{array}{rr} 2768 & 455 \\ 1417 & 3054 \end{array}\right),\left(\begin{array}{rr} 6209 & 32 \\ 6208 & 33 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 3678 & 4235 \end{array}\right),\left(\begin{array}{rr} 3977 & 6214 \\ 3978 & 1429 \end{array}\right)$.
The torsion field $K:=\Q(E[6240])$ is a degree-$309162147840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6240\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $8$ | \( 4225 = 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $18$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 225 = 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 38025bc
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 15a5, its twist by $-195$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{195}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{39}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{10}, \sqrt{78})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{195})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.9253764000000.33 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.2368963584000000.58 | \(\Z/8\Z\) | not in database |
| $8$ | \(\Q(\sqrt{2}, \sqrt{5}, \sqrt{39})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.8.606454677504000000.3 | \(\Z/16\Z\) | not in database |
| $8$ | 8.8.606454677504000000.5 | \(\Z/16\Z\) | 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 | \(\Z/16\Z\) | not in database |
| $16$ | 16.16.367787275866480643670016000000000000.2 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | 13 |
|---|---|---|---|---|
| Reduction type | ord | add | add | add |
| $\lambda$-invariant(s) | ? | - | - | - |
| $\mu$-invariant(s) | ? | - | - | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry ? indicates that the invariants have not yet been computed.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.