Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-41210x+3216100\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-41210xz^2+3216100z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-53408187x+150210586134\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(60, 950)$ | $0.13236234586927368908243994205$ | $\infty$ |
| $(116, -58)$ | $0$ | $2$ |
Integral points
\( \left(-234, 362\right) \), \( \left(-234, -128\right) \), \( \left(-80, 2490\right) \), \( \left(-80, -2410\right) \), \( \left(60, 950\right) \), \( \left(60, -1010\right) \), \( \left(116, -58\right) \), \( \left(120, -10\right) \), \( \left(120, -110\right) \), \( \left(130, 180\right) \), \( \left(130, -310\right) \), \( \left(180, 1190\right) \), \( \left(180, -1370\right) \), \( \left(620, 14390\right) \), \( \left(620, -15010\right) \)
Invariants
| Conductor: | $N$ | = | \( 6370 \) | = | $2 \cdot 5 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $1223549600000$ | = | $2^{8} \cdot 5^{5} \cdot 7^{6} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{65787589563409}{10400000} \) | = | $2^{-8} \cdot 5^{-5} \cdot 7^{3} \cdot 13^{-1} \cdot 73^{3} \cdot 79^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3299059852106669206949015359$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.35695091068301026814222516418$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9795846641235626$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.965310170759242$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.13236234586927368908243994205$ |
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| Real period: | $\Omega$ | ≈ | $0.83515168905070424579686117421$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 160 $ = $ 2^{3}\cdot5\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.4217054687774971023866153320 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.421705469 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.835152 \cdot 0.132362 \cdot 160}{2^2} \\ & \approx 4.421705469\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23040 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $5$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $1$ | $I_{1}$ | split 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 |
|---|---|---|---|
| $2$ | 2B | 4.6.0.3 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1982 & 1561 \\ 231 & 1044 \end{array}\right),\left(\begin{array}{rr} 3119 & 0 \\ 0 & 3639 \end{array}\right),\left(\begin{array}{rr} 848 & 1043 \\ 3325 & 1562 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 28 & 75 \end{array}\right),\left(\begin{array}{rr} 1821 & 1568 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 911 & 1568 \\ 784 & 2633 \end{array}\right),\left(\begin{array}{rr} 3633 & 8 \\ 3632 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[3640])$ is a degree-$811550638080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3640\Z)$.
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 | $4$ | \( 3185 = 5 \cdot 7^{2} \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 1274 = 2 \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 130 = 2 \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 6370bb
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 130c1, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.50960.2 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.10971993760000.19 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.2897292102250000.13 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 |
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 | ord | split | add | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 1 | 2 | - | 1 | 2 | 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 |
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.