Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-6691809x-6663460268\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-6691809xz^2-6663460268z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8672583843x-310864384500642\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(14731/4, 1077595/8)$ | $9.9381505271067429366550415708$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 19110 \) | = | $2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-8288031338496000$ | = | $-1 \cdot 2^{15} \cdot 3^{3} \cdot 5^{3} \cdot 7^{8} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{5748703487739833929}{1437696000} \) | = | $-1 \cdot 2^{-15} \cdot 3^{-3} \cdot 5^{-3} \cdot 7 \cdot 11^{3} \cdot 13^{-1} \cdot 85133^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4299635436881443118276102036$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1326901109846021084240417080$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9985166566477593$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.9609438158528585$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $9.9381505271067429366550415708$ |
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| Real period: | $\Omega$ | ≈ | $0.046933579298419688552525833608$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 9 $ = $ 1\cdot3\cdot1\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.1978967825923617152375545731 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.197896783 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.046934 \cdot 9.938151 \cdot 9}{1^2} \\ & \approx 4.197896783\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 544320 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{15}$ | nonsplit multiplicative | 1 | 1 | 15 | 15 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $7$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
| $13$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $3$ | 3B.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 1557 & 1558 \\ 1550 & 1553 \end{array}\right),\left(\begin{array}{rr} 458 & 1107 \\ 1 & 586 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 391 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1555 & 6 \\ 1554 & 7 \end{array}\right),\left(\begin{array}{rr} 1081 & 6 \\ 123 & 19 \end{array}\right),\left(\begin{array}{rr} 781 & 6 \\ 783 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 937 & 6 \\ 1251 & 19 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$57967902720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $4$ | \( 9555 = 3 \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $3$ | split multiplicative | $4$ | \( 637 = 7^{2} \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 19110r
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 19110g2, its twist by $-7$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.76440.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.223587.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.9115194816000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.149973439707.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.17529220800.6 | \(\Z/6\Z\) | not in database |
| $9$ | 9.1.27898716737855815488000.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.86098815370300115687231251260027000000000000.8 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.2335015186857348704782293796384502034432000000.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.1942732635465314122378868438591905692647424000000000.2 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | split | nonsplit | add | ss | split | ord | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 9 | 2 | 1 | - | 1,1 | 2 | 1 | 1 | 1,1 | 1 | 3 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 0 | 1 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.