Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-1774913x+909412031\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-1774913xz^2+909412031z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2300287275x+42464032035750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(759, 88)$ | $0.64895215791975721858796796461$ | $\infty$ |
| $(3075/4, -3079/8)$ | $0$ | $2$ |
Integral points
\( \left(-1425, 24112\right) \), \( \left(-1425, -22688\right) \), \( \left(759, 88\right) \), \( \left(759, -848\right) \), \( \left(775, -88\right) \), \( \left(775, -688\right) \)
Invariants
| Conductor: | $N$ | = | \( 25350 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $316368000000$ | = | $2^{10} \cdot 3^{2} \cdot 5^{6} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{18013780041269221}{9216} \) | = | $2^{-10} \cdot 3^{-2} \cdot 11^{3} \cdot 23831^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9747248095898195299289310046$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.52876851400738515861517947760$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0923393619522563$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.402219342297952$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.64895215791975721858796796461$ |
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| Real period: | $\Omega$ | ≈ | $0.58915630442532175964089987832$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 80 $ = $ ( 2 \cdot 5 )\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.6466851021768393041959517002 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.646685102 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.589156 \cdot 0.648952 \cdot 80}{2^2} \\ & \approx 7.646685102\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 307200 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $13$ | $2$ | $III$ | additive | -1 | 2 | 3 | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $5$ | 5B | 5.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 780 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 391 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 761 & 20 \\ 760 & 21 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 540 & 431 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 521 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 623 & 760 \\ 0 & 779 \end{array}\right),\left(\begin{array}{rr} 602 & 15 \\ 385 & 518 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[780])$ is a degree-$201277440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/780\Z)$.
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 | $4$ | \( 325 = 5^{2} \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $14$ | \( 507 = 3 \cdot 13^{2} \) |
| $13$ | additive | $50$ | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 25350cc
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 1014c4, its twist by $5$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.7909200.2 | \(\Z/4\Z\) | not in database |
| $4$ | 4.4.274625.1 | \(\Z/10\Z\) | not in database |
| $8$ | 8.4.111209679360000.15 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.62555444640000.74 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | 8.8.75418890625.1 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/20\Z\) | not in database |
| $20$ | 20.0.5253278865130934441704266071319580078125.4 | \(\Z/2\Z \oplus \Z/10\Z\) | 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 | nonsplit | add | ord | ss | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 1 | - | 3 | 3,1 | - | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 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.