Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-x^2-83x+318\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-x^2z-83xz^2+318z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-108000x+13554000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2, 12)$ | $0.31631685960117170637616532578$ | $\infty$ |
Integral points
\( \left(2, 12\right) \), \( \left(2, -13\right) \), \( \left(6, 0\right) \), \( \left(6, -1\right) \), \( \left(158, 1976\right) \), \( \left(158, -1977\right) \)
Invariants
| Conductor: | $N$ | = | \( 925 \) | = | $5^{2} \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $578125$ | = | $5^{6} \cdot 37 $ |
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| j-invariant: | $j$ | = | \( \frac{4096000}{37} \) | = | $2^{15} \cdot 5^{3} \cdot 37^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.071811772927356149088080564782$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.87653072914440633638846023139$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8826782160855301$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6431772139239973$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.31631685960117170637616532578$ |
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| Real period: | $\Omega$ | ≈ | $2.9208099320522509910335146196$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.8478028503973594964313234745 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.847802850 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.920810 \cdot 0.316317 \cdot 2}{1^2} \\ & \approx 1.847802850\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 96 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $37$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $3$ | 3B | 27.36.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9990 = 2 \cdot 3^{3} \cdot 5 \cdot 37 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 31 & 36 \\ 4168 & 3229 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6251 & 1275 \\ 8965 & 7306 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right),\left(\begin{array}{rr} 6556 & 4005 \\ 9005 & 4786 \end{array}\right),\left(\begin{array}{rr} 5993 & 0 \\ 0 & 9989 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 9937 & 54 \\ 9936 & 55 \end{array}\right)$.
The torsion field $K:=\Q(E[9990])$ is a degree-$1275231651840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9990\Z)$.
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 |
|---|---|---|---|
| $5$ | additive | $14$ | \( 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 25 = 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 925b
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 37b3, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/3\Z\) | 2.2.5.1-1369.1-a2 |
| $3$ | 3.3.148.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.810448.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.6325293375.3 | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.234270125.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.6.2738000.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.21347865140625.1 | \(\Z/9\Z\) | not in database |
| $12$ | 12.12.10262905636000000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.1419075219912557986902228696000000000.2 | \(\Z/6\Z\) | not in database |
| $18$ | 18.18.72096490367960066397512000000000.1 | \(\Z/18\Z\) | 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 | ss | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | nonsplit | ord | ord | ord |
| $\lambda$-invariant(s) | 2,7 | 1 | - | 1 | 1 | 1 | 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.