Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-39838x-3045708\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-39838xz^2-3045708z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-51630075x-141945662250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-118, -66)$ | $0.92666342635567140718430550330$ | $\infty$ |
| $(-122, 12)$ | $1.8029999184383212451728634070$ | $\infty$ |
| $(-108, 54)$ | $0$ | $2$ |
Integral points
\( \left(-122, 110\right) \), \( \left(-122, 12\right) \), \( \left(-118, 184\right) \), \( \left(-118, -66\right) \), \( \left(-108, 54\right) \), \( \left(242, 1104\right) \), \( \left(242, -1346\right) \), \( \left(292, 3054\right) \), \( \left(292, -3346\right) \), \( \left(382, 5934\right) \), \( \left(382, -6316\right) \), \( \left(1026, 31680\right) \), \( \left(1026, -32706\right) \), \( \left(2132, 96934\right) \), \( \left(2132, -99066\right) \), \( \left(2596, 130574\right) \), \( \left(2596, -133170\right) \), \( \left(7298, 619582\right) \), \( \left(7298, -626880\right) \)
Invariants
| Conductor: | $N$ | = | \( 31850 \) | = | $2 \cdot 5^{2} \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $47794906250000$ | = | $2^{4} \cdot 5^{9} \cdot 7^{6} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{3803721481}{26000} \) | = | $2^{-4} \cdot 5^{-3} \cdot 7^{3} \cdot 13^{-1} \cdot 223^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4588172888883155797452698646$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.31885674185639126010778617373$ |
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| $abc$ quality: | $Q$ | ≈ | $0.906187683689355$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.184801022561549$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.6160655954603847739218943651$ |
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| Real period: | $\Omega$ | ≈ | $0.33806795511413323307984398688$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $8.7414398590015416849905458499 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.741439859 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.338068 \cdot 1.616066 \cdot 64}{2^2} \\ & \approx 8.741439859\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 165888 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $7$ | $4$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.3 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 24 \\ 492 & 1687 \end{array}\right),\left(\begin{array}{rr} 3119 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 8191 & 4704 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 4 \\ 10820 & 10901 \end{array}\right),\left(\begin{array}{rr} 5461 & 4704 \\ 5460 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 10897 & 24 \\ 10896 & 25 \end{array}\right),\left(\begin{array}{rr} 3256 & 21 \\ 7035 & 4306 \end{array}\right),\left(\begin{array}{rr} 7281 & 6244 \\ 5740 & 4761 \end{array}\right),\left(\begin{array}{rr} 1854 & 10913 \\ 1351 & 9400 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$4869303828480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 | $4$ | \( 15925 = 5^{2} \cdot 7^{2} \cdot 13 \) |
| $5$ | additive | $18$ | \( 1274 = 2 \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 650 = 2 \cdot 5^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 31850ce
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 130a1, its twist by $-35$.
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 |
| $2$ | \(\Q(\sqrt{-35}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.50960.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-35}, \sqrt{65})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.529006842000.2 | \(\Z/6\Z\) | 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 |
| $8$ | 8.0.64923040000.14 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.99632675735549813495284242187500000000.2 | \(\Z/18\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | ord | add | add | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 5 | 6 | - | - | 2 | 3 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | 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.