Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-1616855x+4941345647\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-1616855xz^2+4941345647z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-25869675x+316220251750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1479, 75310)$ | $0.28178428943106902288824036919$ | $\infty$ |
Integral points
\( \left(-1901, 34750\right) \), \( \left(-1901, -32850\right) \), \( \left(579, 64510\right) \), \( \left(579, -65090\right) \), \( \left(1479, 75310\right) \), \( \left(1479, -76790\right) \), \( \left(22435, 3344446\right) \), \( \left(22435, -3366882\right) \)
Invariants
| Conductor: | $N$ | = | \( 76050 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-10275857780123520000000$ | = | $-1 \cdot 2^{13} \cdot 3^{9} \cdot 5^{7} \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{50308609}{1105920} \) | = | $-1 \cdot 2^{-13} \cdot 3^{-3} \cdot 5^{-1} \cdot 13 \cdot 157^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9044808680501103833351789591$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.15951047080868580703181495369$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9850859312951057$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.175196357819349$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.28178428943106902288824036919$ |
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| Real period: | $\Omega$ | ≈ | $0.10798715721032120738819863003$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 312 $ = $ 13\cdot2\cdot2^{2}\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.4938743210037490073347631937 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.493874321 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.107987 \cdot 0.281784 \cdot 312}{1^2} \\ & \approx 9.493874321\end{aligned}$$
Modular invariants
Modular form 76050.2.a.dy
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4672512 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $13$ | $I_{13}$ | split multiplicative | -1 | 1 | 13 | 13 |
| $3$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $13$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 31 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 119 & 2 \\ 118 & 3 \end{array}\right),\left(\begin{array}{rr} 61 & 2 \\ 61 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 119 & 0 \end{array}\right),\left(\begin{array}{rr} 97 & 2 \\ 97 & 3 \end{array}\right),\left(\begin{array}{rr} 41 & 2 \\ 41 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$17694720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
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 | $4$ | \( 38025 = 3^{2} \cdot 5^{2} \cdot 13^{2} \) |
| $3$ | additive | $6$ | \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $18$ | \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $74$ | \( 225 = 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 76050en consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 5070d1, its twist by $-195$.
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 |
|---|---|---|---|
| $3$ | 3.1.20280.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.49353408000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.15615726750000.4 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | add | add | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | - | - | 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 |
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.