Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-313880x+67796997\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-313880xz^2+67796997z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5022075x+4333985750\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 13950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 31$ |
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| Discriminant: | $\Delta$ | = | $-1956623538656250$ | = | $-1 \cdot 2 \cdot 3^{7} \cdot 5^{6} \cdot 31^{5} $ |
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| j-invariant: | $j$ | = | \( -\frac{300238092661681}{171774906} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 29^{3} \cdot 31^{-5} \cdot 2309^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8803654106442210770372032760$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.52634031009311604403920099093$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0100188200478981$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.195815400176143$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.46142540160108103384348345895$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 10 $ = $ 1\cdot2\cdot1\cdot5 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.6142540160108103384348345895 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 4.614254016 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.461425 \cdot 1.000000 \cdot 10}{1^2} \\ & \approx 4.614254016\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 112000 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $31$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
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 |
|---|---|---|
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 2783 & 1850 \\ 40 & 1359 \end{array}\right),\left(\begin{array}{rr} 1801 & 10 \\ 1565 & 51 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 3665 & 3601 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 2479 & 3710 \\ 1235 & 3669 \end{array}\right),\left(\begin{array}{rr} 1861 & 10 \\ 1865 & 51 \end{array}\right),\left(\begin{array}{rr} 2791 & 10 \\ 2795 & 51 \end{array}\right),\left(\begin{array}{rr} 3711 & 10 \\ 3710 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3720])$ is a degree-$658243584000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3720\Z)$.
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 | $4$ | \( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \) |
| $3$ | additive | $8$ | \( 1550 = 2 \cdot 5^{2} \cdot 31 \) |
| $5$ | additive | $14$ | \( 18 = 2 \cdot 3^{2} \) |
| $31$ | split multiplicative | $32$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 13950.cv
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 186.c1, its twist by $-15$.
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 |
|---|---|---|---|
| $3$ | 3.1.744.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\zeta_{15})^+\) | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.411830784.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $10$ | 10.0.246037500000000.1 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | 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 | add | add | ord | ord | ord | ord | ord | ord | ss | split | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | - | - | 0 | 0 | 2 | 0 | 0 | 0 | 0,0 | 1 | 0 | 0 | 0 | 0 |
| $\mu$-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
All $p$-adic regulators are identically $1$ since the rank is $0$.