Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+245x-975\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+245xz^2-975z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+317493x-46442106\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(50, 345\right) \) | $0$ | $6$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([50:345:1]\) | $0$ | $6$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1803, 79920\right) \) | $0$ | $6$ |
Integral points
\( \left(10, 45\right) \), \( \left(10, -55\right) \), \( \left(50, 345\right) \), \( \left(50, -395\right) \)
\([10:45:1]\), \([10:-55:1]\), \([50:345:1]\), \([50:-395:1]\)
\((363,\pm 10800)\), \((1803,\pm 79920)\)
Invariants
| Conductor: | $N$ | = | \( 370 \) | = | $2 \cdot 5 \cdot 37$ |
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| Minimal Discriminant: | $\Delta$ | = | $-1369000000$ | = | $-1 \cdot 2^{6} \cdot 5^{6} \cdot 37^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{1625964918479}{1369000000} \) | = | $2^{-6} \cdot 5^{-6} \cdot 11^{3} \cdot 37^{-2} \cdot 1069^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.44077519776586893049499454884$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.44077519776586893049499454884$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9481798185675541$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.7547320978211705$ | |||
| 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: | $\Omega$ | ≈ | $0.84030079696647370954699451897$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ ( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.6806015939329474190939890379 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.680601594 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.840301 \cdot 1.000000 \cdot 72}{6^2} \\ & \approx 1.680601594\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 192 |
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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 semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $5$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $37$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | 4.12.0.12 | $12$ |
| $3$ | 3B.1.1 | 3.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4440 = 2^{3} \cdot 3 \cdot 5 \cdot 37 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 1695 & 22 \\ 1862 & 443 \end{array}\right),\left(\begin{array}{rr} 1111 & 24 \\ 555 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 24 \\ 3732 & 3133 \end{array}\right),\left(\begin{array}{rr} 3559 & 18 \\ 2298 & 3499 \end{array}\right),\left(\begin{array}{rr} 4417 & 24 \\ 4416 & 25 \end{array}\right),\left(\begin{array}{rr} 2979 & 16 \\ 886 & 123 \end{array}\right),\left(\begin{array}{rr} 2221 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 314 & 335 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[4440])$ is a degree-$167931740160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4440\Z)$.
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 | $4$ | \( 1 \) |
| $3$ | good | $2$ | \( 37 \) |
| $5$ | split multiplicative | $6$ | \( 74 = 2 \cdot 37 \) |
| $37$ | split multiplicative | $38$ | \( 10 = 2 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 370d
consists of 4 curves linked by isogenies of
degrees dividing 6.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | 2.0.4.1-68450.5-h6 |
| $4$ | \(\Q(\sqrt{26 +2 \sqrt{185}})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{13 +4 i})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $6$ | 6.0.50602347.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.299865760000.8 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $9$ | 9.3.50602347000000.1 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $18$ | 18.0.10488207449736843264000000000000.2 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 37 |
|---|---|---|---|---|
| Reduction type | split | ord | split | split |
| $\lambda$-invariant(s) | 2 | 2 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.