Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+2305x-15975\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+2305xz^2-15975z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+2987253x-754291386\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{7}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(10, 85\right) \) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([10:85:1]\) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(363, 19440\right) \) | $0$ | $7$ |
Integral points
\( \left(10, 85\right) \), \( \left(10, -95\right) \), \( \left(40, 355\right) \), \( \left(40, -395\right) \), \( \left(190, 2605\right) \), \( \left(190, -2795\right) \)
\([10:85:1]\), \([10:-95:1]\), \([40:355:1]\), \([40:-395:1]\), \([190:2605:1]\), \([190:-2795:1]\)
\((363,\pm 19440)\), \((1443,\pm 81000)\), \((6843,\pm 583200)\)
Invariants
| Conductor: | $N$ | = | \( 1230 \) | = | $2 \cdot 3 \cdot 5 \cdot 41$ |
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| Minimal Discriminant: | $\Delta$ | = | $-896670000000$ | = | $-1 \cdot 2^{7} \cdot 3^{7} \cdot 5^{7} \cdot 41 $ |
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| j-invariant: | $j$ | = | \( \frac{1354330706847119}{896670000000} \) | = | $2^{-7} \cdot 3^{-7} \cdot 5^{-7} \cdot 31^{3} \cdot 41^{-1} \cdot 43^{3} \cdot 83^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.98228345427979430621940970805$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.98228345427979430621940970805$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9941892399139366$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.897148675973694$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $0.50464667342472116309390240809$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 343 $ = $ 7\cdot7\cdot7\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $7$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.5325267139730481416573168567 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.532526714 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.504647 \cdot 1.000000 \cdot 343}{7^2} \\ & \approx 3.532526714\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1960 |
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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 semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $3$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $5$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $41$ | $1$ | $I_{1}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $7$ | 7B.1.1 | 7.48.0.1 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 34440 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 41 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 8611 & 17234 \\ 0 & 33211 \end{array}\right),\left(\begin{array}{rr} 26041 & 14 \\ 10087 & 99 \end{array}\right),\left(\begin{array}{rr} 34427 & 14 \\ 34426 & 15 \end{array}\right),\left(\begin{array}{rr} 13777 & 14 \\ 27559 & 99 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 22961 & 14 \\ 22967 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 25831 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 17221 & 14 \\ 17227 & 99 \end{array}\right)$.
The torsion field $K:=\Q(E[34440])$ is a degree-$2047604686848000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/34440\Z)$.
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 | $4$ | \( 615 = 3 \cdot 5 \cdot 41 \) |
| $3$ | split multiplicative | $4$ | \( 410 = 2 \cdot 5 \cdot 41 \) |
| $5$ | split multiplicative | $6$ | \( 246 = 2 \cdot 3 \cdot 41 \) |
| $7$ | good | $2$ | \( 41 \) |
| $41$ | nonsplit multiplicative | $42$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 1230k
consists of 2 curves linked by isogenies of
degree 7.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.4920.1 | \(\Z/14\Z\) | not in database |
| $6$ | 6.0.119095488000.1 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $8$ | 8.2.5005750838670000.5 | \(\Z/21\Z\) | not in database |
| $12$ | deg 12 | \(\Z/28\Z\) | 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 | 41 |
|---|---|---|---|---|---|
| Reduction type | split | split | split | ord | nonsplit |
| $\lambda$-invariant(s) | 2 | 1 | 1 | 4 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.