Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-2116482x+1183671476\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-2116482xz^2+1183671476z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2742961347x+55266520801086\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(812, 734)$ | $2.1348996132898116467510259881$ | $\infty$ |
| $(3419/4, -3419/8)$ | $0$ | $2$ |
Integral points
\( \left(812, 734\right) \), \( \left(812, -1546\right) \)
Invariants
| Conductor: | $N$ | = | \( 102410 \) | = | $2 \cdot 5 \cdot 7^{2} \cdot 11 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $601923833697032000$ | = | $2^{6} \cdot 5^{3} \cdot 7^{6} \cdot 11^{6} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{8912089320684236569}{5116268168000} \) | = | $2^{-6} \cdot 5^{-3} \cdot 11^{-6} \cdot 19^{-2} \cdot 23^{3} \cdot 109^{3} \cdot 827^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3575701116627576313720834256$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3846150371351009788194070539$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9673933417704673$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.794196886215084$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.1348996132898116467510259881$ |
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| Real period: | $\Omega$ | ≈ | $0.28627917904427116904055506660$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2\cdot3\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.3341277036144712275330875681 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.334127704 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.286279 \cdot 2.134900 \cdot 48}{2^2} \\ & \approx 7.334127704\end{aligned}$$
Modular invariants
Modular form 102410.2.a.z
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2903040 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $19$ | $2$ | $I_{2}$ | nonsplit 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 | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 87780 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 50159 & 0 \\ 0 & 87779 \end{array}\right),\left(\begin{array}{rr} 42855 & 38668 \\ 17738 & 1037 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 87730 & 87771 \end{array}\right),\left(\begin{array}{rr} 87769 & 12 \\ 87768 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 60061 & 37632 \\ 71946 & 50233 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 71821 & 37632 \\ 54726 & 50233 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 29261 & 37632 \\ 29260 & 1 \end{array}\right),\left(\begin{array}{rr} 50170 & 75243 \\ 77721 & 37612 \end{array}\right)$.
The torsion field $K:=\Q(E[87780])$ is a degree-$75487586549760000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/87780\Z)$.
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 | $4$ | \( 245 = 5 \cdot 7^{2} \) |
| $3$ | good | $2$ | \( 931 = 7^{2} \cdot 19 \) |
| $5$ | split multiplicative | $6$ | \( 20482 = 2 \cdot 7^{2} \cdot 11 \cdot 19 \) |
| $7$ | additive | $26$ | \( 2090 = 2 \cdot 5 \cdot 11 \cdot 19 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 9310 = 2 \cdot 5 \cdot 7^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 102410.z
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 2090.c1, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ 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 |
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.4.171229520.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{21})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.402300927.1 | \(\Z/6\Z\) | 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 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 |
| $18$ | 18.6.16402017354713010830231835423946864784851149000000000000.1 | \(\Z/18\Z\) | 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 | ord | split | add | nonsplit | ord | ss | nonsplit | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 5 | 1 | 2 | - | 1 | 1 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 3 |
| $\mu$-invariant(s) | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.