Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-18136x-940480\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-18136xz^2-940480z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-23504283x-43808522058\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-76, 56)$ | $0.89116541819962799126126719029$ | $\infty$ |
| $(-80, 40)$ | $0$ | $2$ |
Integral points
\( \left(-80, 40\right) \), \( \left(-76, 56\right) \), \( \left(-76, 20\right) \), \( \left(176, 1064\right) \), \( \left(176, -1240\right) \), \( \left(704, 17960\right) \), \( \left(704, -18664\right) \)
Invariants
| Conductor: | $N$ | = | \( 21090 \) | = | $2 \cdot 3 \cdot 5 \cdot 19 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $895635947520$ | = | $2^{20} \cdot 3^{5} \cdot 5 \cdot 19 \cdot 37 $ |
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| j-invariant: | $j$ | = | \( \frac{659704930833045889}{895635947520} \) | = | $2^{-20} \cdot 3^{-5} \cdot 5^{-1} \cdot 11^{3} \cdot 19^{-1} \cdot 37^{-1} \cdot 79139^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1985249098003686275019811138$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1985249098003686275019811138$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9293969022266227$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.1209607187885915$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.89116541819962799126126719029$ |
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| Real period: | $\Omega$ | ≈ | $0.41143332011618471770228604689$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 100 $ = $ ( 2^{2} \cdot 5 )\cdot5\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.1663786695650292437279403464 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.166378670 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.411433 \cdot 0.891165 \cdot 100}{2^2} \\ & \approx 9.166378670\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 41600 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $20$ | $I_{20}$ | split multiplicative | -1 | 1 | 20 | 20 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $37$ | $1$ | $I_{1}$ | nonsplit 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 | 8.12.0.12 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 84360 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \cdot 37 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 84354 & 84355 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 31643 & 31638 \\ 52730 & 10547 \end{array}\right),\left(\begin{array}{rr} 22208 & 3 \\ 66605 & 2 \end{array}\right),\left(\begin{array}{rr} 84353 & 8 \\ 84352 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 28124 & 1 \\ 56263 & 6 \end{array}\right),\left(\begin{array}{rr} 33748 & 1 \\ 67511 & 6 \end{array}\right),\left(\begin{array}{rr} 10553 & 10548 \\ 10550 & 52727 \end{array}\right),\left(\begin{array}{rr} 22808 & 3 \\ 77525 & 2 \end{array}\right)$.
The torsion field $K:=\Q(E[84360])$ is a degree-$165406046787993600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/84360\Z)$.
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 | $4$ | \( 10545 = 3 \cdot 5 \cdot 19 \cdot 37 \) |
| $3$ | split multiplicative | $4$ | \( 7030 = 2 \cdot 5 \cdot 19 \cdot 37 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 703 = 19 \cdot 37 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 21090.n
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-703}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{10545}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{-703})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | split | nonsplit | ss | ss | ord | ord | nonsplit | ord | ord | ss | nonsplit | ord | ord | ss |
| $\lambda$-invariant(s) | 2 | 2 | 5 | 1,1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 3,1 | 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 | 0,0 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.