Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-18594x-2460780\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-18594xz^2-2460780z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-297507x-157787426\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(18081/64, 1895733/512)$ | $9.0031168311868354867545139002$ | $\infty$ |
| $(180, -90)$ | $0$ | $2$ |
Integral points
\( \left(180, -90\right) \)
Invariants
| Conductor: | $N$ | = | \( 16830 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-2214397134766080$ | = | $-1 \cdot 2^{18} \cdot 3^{12} \cdot 5 \cdot 11 \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{975276594443809}{3037581803520} \) | = | $-1 \cdot 2^{-18} \cdot 3^{-6} \cdot 5^{-1} \cdot 7^{3} \cdot 11^{-1} \cdot 17^{-2} \cdot 31^{3} \cdot 457^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6310786010646686056727866934$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0817724567306137599751640749$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9490279432038693$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.414680779673124$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $9.0031168311868354867545139002$ |
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| Real period: | $\Omega$ | ≈ | $0.18865392209764605569965502179$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.3969466026134545538271789087 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.396946603 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.188654 \cdot 9.003117 \cdot 8}{2^2} \\ & \approx 3.396946603\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 110592 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{18}$ | nonsplit multiplicative | 1 | 1 | 18 | 18 |
| $3$ | $2$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 22440 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11221 & 12 \\ 6 & 73 \end{array}\right),\left(\begin{array}{rr} 14521 & 12 \\ 19806 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 4461 & 22432 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 22429 & 12 \\ 22428 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6555 & 10288 \\ 6518 & 10277 \end{array}\right),\left(\begin{array}{rr} 4090 & 3 \\ 20373 & 22432 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 22390 & 22431 \end{array}\right),\left(\begin{array}{rr} 7471 & 22438 \\ 18642 & 22427 \end{array}\right)$.
The torsion field $K:=\Q(E[22440])$ is a degree-$381186736128000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/22440\Z)$.
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 | $4$ | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
| $3$ | additive | $2$ | \( 935 = 5 \cdot 11 \cdot 17 \) |
| $5$ | split multiplicative | $6$ | \( 3366 = 2 \cdot 3^{2} \cdot 11 \cdot 17 \) |
| $11$ | split multiplicative | $12$ | \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 16830bh
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 5610be1, its twist by $-3$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-55}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/6\Z\) | not in database |
| $3$ | 3.1.2622675.2 | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.9155520.3 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-55})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.20635272466875.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.378313328559375.1 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.3247089749841984252193978896928313671875.2 | \(\Z/18\Z\) | 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 | split | ord | split | ord | split | ord | ord | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 6 | - | 2 | 1 | 2 | 3 | 2 | 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 |
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.